Obsidian Vault

Notes 10-15

Sat, 17 Oct 2026 18:49:56 GMT

We learned about Fourier Transform, not to be confused to Fourier SignalsThe Fourier Signal of a Signal is composed of an discretely infinite series of orthogonal functions
The transform can be used generally for any Periodic Signal and is composed of a generally infinite series of orthogonal functions
Periodic Signal
Aperiodic Signals can be converted into Periodic Signals via periodization
Periodization
The existence conditions of a Fourier Transform has both a weak and strong condition, only the strong condition guarantees for all
Fourier Transform
The properties of Fourier transforms generally follow the properties of a Fourier Signal
Fourier Transform
We also went over common Fourier Transforms
Fourier Transform of Common Signals

Fourier Transform of Common Signals

Sat, 17 Oct 2026 18:45:03 GMT

The magnitude of the Fourier Transform of is the same as the magnitude spectrum of
The angle of the Fourier transform of is the same as the the phase spectrum of
The Fourier transform of the Impulse Function The impulse function has a very narrow width in the time domainY axisX axis00-10-10-5-5551010-5-555Expression 1But a large width in the frequency domainY axisX axis00-10-10-8-8-6-6-4-4-2-2224466881010-6-6-4-4-2-2224466Expression 1The inverse Fourier transform of The Fourier transform of the unit step function with an exponentialWhere the magnitude of is given by Y axisX axis00-10-10-5-5551010-5-555Expression 1And the phase of is given by Y axisX axis00-10-10-5-5551010-5-555Expression 1Consider function Where can be separatedConsider a time window functionY axisX axis00-10-10-5-5551010-5-555Expression 1
The period can be described using the Sinc FunctionConsider the same time window function in the frequency domainY axisX axis00-10-10-5-5551010-5-555Expression 1Using Euler's formula and multiplying by the term Such that the domain of the inverse Fourier transform can be given as

Periodization

Sat, 17 Oct 2026 18:41:09 GMT

Periodization is a method of converting Aperiodic Signals into Periodic SignalsConsider a signalY axisX axis00-10-10-5-5551010-5-555Expression 1Construct an a periodic signal by repeating over a given Y axisX axis00-10-10-5-5551010-5-555Expression 1

Fourier Transform

Sat, 17 Oct 2026 18:40:54 GMT

A Fourier transform on signal exists when is an Energy SignalThen the Fourier of Would be finiteAnd the energy signal of the error would be minimizedHowever, and may differ at some point A Fourier transform on signal exists whenAnd has a finite number of maxima within any finite intervalAnd has a finite number of discontinuities within any finite intervalThen for all values of where is continuousAt a discontinuity point For real

Fourier Series of Exponential Function

Fri, 16 Oct 2026 00:23:40 GMT

For a complex exponential set ofThe complex exponential set is orthogonalSince The complex exponential set is CompleteFor a periodic signal with a period of and For real For a real valued signal The nth term of the Trigonometric SeriesRecall that The nth term of the Exponential SeriesFollowing the formSince the original signal is real has to be the complex conjugate of Define such thatWhere is periodic with period So that the the derivative of the convolutionDefine as a Fourier of exponentialsBy rearranging the integrals,The function is real if
For a real SignalBy equating the magnitudes of the signalsWe can see that the magnitude is evenBy equating the angles of the signalsWe can see that the angle is oddConsider the Fourier seriesThe power the harmonic of can be determined bySo the power of the signal can be determined byUse the conjugate identity for the magnitude of a signalSubstitute the Fourier series of Since By equating to the inner productWhich is when , so that If is real and even Then is real and evenIf is real and odd Then is imaginary and oddThe even component of a real The odd component of a real

Fourier Series of Trigonometric Functions

Fri, 16 Oct 2026 00:05:13 GMT

The trigonometric set can be defined asWhere it is orthogonal over any interval of duration The set is defined as CompleteSo that an arbitrary function can be defined as a sum of elements within the trigonometric setThe total energy of the trigonometric set is with equating to Each term can be defined with respect to its energyIf is evenIf is oddConsider the function
The period of the signal is
The signal also exhibits as an Even Signal so that So that the function can be defined asSince the trigonometric components of a function can be compacted Since ,
This form is Periodic so that any function that can be represented over one period can be represented across all

Sinc Function

Fri, 16 Oct 2026 00:04:45 GMT

Define a function such thatY axisX axis00-10-10-5-5551010-5-555Expression 1The function can be used to describe time windows within the frequency domain

Aperiodic Signal

Thu, 15 Oct 2026 23:25:48 GMT

A Signal is aperiodic when it does not repeat

Periodic Signal

Thu, 15 Oct 2026 23:24:52 GMT

A Signal is periodic if it repeats over a period indefinitely
The Fourier Series Representation of Periodic Signals can be given byWhereFor signals with large periods such that Where is defined asSo thatAs the sum becomes an integral in respect to
This is the basis of the Fourier Transform

Fourier Series of Periodic Signals

Thu, 15 Oct 2026 23:11:33 GMT

Signals can be represented by a set of orthogonal signalsOn real valued signals over a finite interval The inner product is defined asFor an orthogonal signal set within the same interval The inner product is defined asSo that when the energy of the signal is The signal can be approximatedWhere is a coefficient of approximation over all set of signalsFor any given signal can be chosen to minimize the error between the approximation andThis can be done by defining the error signalAnd minimizing the energy of that error signalSinceSimplifyingWe can expand out the square on the third termSo thatResulting in We can then find the minimum energyIf the error signal approaches 0 as approaches infinity then that signal is CompleteConsider two periodic signalsDefine Suppose In order for So is only periodic if is rational If so, and are harmonically relatedMultiple Signals The period of a sum of multiple periodic signals is the least common multiple of the periods of those signals For a real valued signalBut for a complex valued signalSo that is a orthogonal set if

Diode

Thu, 15 Oct 2026 16:22:19 GMT

A diode is an active device that operates on the principles of the P-N JunctionThe bias of the diode determines how voltage is able to move across the diodeIn forward bias, the potential barrier is lowered In reverse bias, the potential barrier is increased Therefore, it is harder for a voltage to pass through a diode in the reverse bias directionThe width of the transition region is determined by and is therefore affected by biasIn forward bias, the width is decreased In reverse bias, the width is increased
The Fermi Level is split in the quasi-neutral region under biasIn forward bias, the fermi separation is decreased In reverse bias, the fermi separation is increased Both carriers have to overcome the potential energy barrier in order to diffuse across the barrierIn forward bias, the barrier is lower so the current is increased In reverse bias, the barrier is higher so the current is decreased
Current within a diode is made up both diffusion current and Drift CurrentIdeally, for The diffusion current is mostly made up of majority carriers while the drift (generation) current is mostly made up of minority carriersSo that the total current can be give byIn reverse bias, when the reverse voltage reaches a critical level, breakdown happensA narrow based diode is when the length of the recombination region is less then the length of the the doped regionsIn long exponential decay For narrow exponential decay For narrow exponential decay, the decay of excess carriers can be though of as linearThe total injected excess carriers is the area of the excess carrier distribution functionWhere the diffusion current can be writtenas

Depletion Region

Thu, 15 Oct 2026 15:48:24 GMT

Half Wave Symmetry

Wed, 14 Oct 2026 00:17:54 GMT

The two halves of one period of a signal are exactly negatives of each otherConsider the function belowY axisX axis00-10-10-5-5551010-5-555Expression 1Expression 2Expression 3Expression 4For half wave symmetryRecall that when is odd and when is even So that for odd harmonicsSolving for By substitutingSince and Consider a signal such that is real is periodic with period for has a Fourier equivalence which is odd Find Find Since is then real and odd, is imaginary and oddSo thatSinceSince is evenBy finding the power of the signal

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Exam 1 Review Example_2

Tue, 13 Oct 2026 19:07:40 GMT

Exam 1 Review Example_1

Tue, 13 Oct 2026 19:00:27 GMT

Exam 1 Review Example_0

Tue, 13 Oct 2026 18:59:31 GMT

Inductor

Tue, 13 Oct 2026 02:49:26 GMT

An inductor is an element that stores the inertia of current via a magnetic fieldThe inductor is given by this symbol The voltage of an inductor is defined byThe current byThe energy of the inductor can be given byInductors, similar to Resistors, combine inductance when in series but split inductance in parallelBut this is only the face if they're magnetically isolated from each other.

Inductors in DC Steady State have no change in current, and therefore no voltage differenceAn inductor in steady state acts as a short

Inductor Example_0

Mon, 12 Oct 2026 16:42:25 GMT

Inductor

Mon, 12 Oct 2026 16:41:51 GMT

Inductors are a circuit component that uses Magnetic Fields to buffer current
During the energizing phase, the inductor creates and sustain a magnetic field from the flow of CurrentDuring the energizing phase, the created magnetic field pushes currentThe inductor is given by this symbol The inductance of a inductor is measured in Henrys, the inductance of a inductor can be determined by a color code systemThe voltage of an inductor is equal to its inductance and the rate of change of the currentThe density of an inductor is equal to its inductance and the current
All Electrical Fields induce a Magnetic Field Orthogonally
Inductors, similar to Resistors, combine inductance when in series but split inductance in parallelBut this is only the face if they're magnetically isolated from each other.
Like Capacitors in a RC circuit, inductors have a phase in which the buffer builds upHowever, unlike capacitors, inductors deal with current, not Voltage Current through an energizing inductor Voltage across the inductor Voltage across the resistor Notice the L/R in the Voltage across the resistor Current through a deenergizing inductor Voltage across the inductor Voltage across the resistor Now notice the R/L in the Voltage across the resistor

Capacitor

Mon, 12 Oct 2026 16:41:02 GMT

A capacitor is an element that stores a chargeThe capacitor is given by this symbolThe current of a capacitor is determined byContinuous Voltage The voltage is assumed continuous since a jump in the voltage would induce an infinite current And the voltage determined by
Capacitors in DC Steady State have no change in voltage, and therefore no current flow
Since the capacitor has no current and an open voltage, it can be modeled as an Thevenin Open Circuit under steady state
Capacitors satisfies the requirements for a Linear Circuit with the addition of initial values of of the capacitor

Unlike, Resistors, capacitors in series divide all voltage across all capacitorsVoltage and Current While resistors in parallel share a voltage split current, capacitors in series will usually share a common current and split voltage Capacitors in parallel combine their capacitance
The equivalent Capacitance for capacitors in parallel can be calculated via the net current, the equivalent capacitance for capacitors in series can be calculated via the net voltage
Consider a circuit with a switch, at time the switch is opened

Capacitor Example_1

Mon, 12 Oct 2026 16:37:57 GMT

Capacitor Example_0

Mon, 12 Oct 2026 16:18:04 GMT

Capacitor Example

Mon, 12 Oct 2026 16:01:50 GMT

Poisson Distribution

Mon, 12 Oct 2026 15:01:09 GMT

The Poisson Distribution, which means fish in French, is a distribution for a Stochastic process represented by only one parameter: its Expected Value For a random variable with the poisson distribution
Unlike a Normal Distribution, the poisson distribution is not symmetric along its expected value, instead it has a tail towards

Binomial Random Variable

Sun, 11 Oct 2026 20:46:46 GMT

The binomial random variable is Random Variable of events with probability
The Probability Mass Function of a binomial random variable with probability isSo that for all events

Bernoulli Random Variable

Sun, 11 Oct 2026 20:03:28 GMT

The Bernoulli random variable takes a certain value depending on the outcome of an EventFor example, the Bernoulli random variable of a coin with probability of flipping a heads is
So that the Probability Mass Function isThe Bernoulli random variable is generally used to model generic probability situations with two outcomes

Probability Mass Function

Sun, 11 Oct 2026 19:59:39 GMT

One of the most important ways of characterizing a Random Variable is the probability mass functionWhere the probability mass is denoted by which is equal to the probability of the event which consists of all outcome that equals For example, let be the number of heads flipped by a fair coin in 2 independent tossesThe probability mass of flipping two heads can be given byAnd more generally

Total Probability Theorem

Sun, 11 Oct 2026 19:49:44 GMT

For disjoint events that form a partition of the sample space , then for any event

Law of Conditional Probability

Sun, 11 Oct 2026 19:47:07 GMT

The Law of Conditional Probability can be used to find the probability that an event will occur given event has occurred which can be written as The law as followsThe law only works when Two coins are flipped, what is the probability that both coins lands on heads given that one coin has landed on headsThe sample space is as followsThe original probability of both coins landing heads is one out of four outcomes or If one coin lands head, the sample space then becomesThe probability then becomes one out of two outcomes or This is given by the law of conditional probability as soWhereThe because we're in the original sample space , the event of one heads and either heads of tails and only heads only share one outcome out of four ThereforeA radar is accurate such that, if an aircraft is present in a certain area, the probability that the radar registers its presence is If an aircraft is not present in a certain area, the probability that the radar registers an aircraft is Assuming the aircraft is present in any given area with a probability of Let the events, and The intersection of conditioning events with positive probabilities can be written as

Node Voltage Analysis

Sun, 11 Oct 2026 17:43:26 GMT

Node Voltage Analysis relies on analyzing the voltages of nodes rather than of Elements which significantly reduces the amount of EquationsConsider this circuitWe can define the bottom-most node as the ground nodeThis means that node is simply defined as which leaves us with two unknown voltages, and
Using KCL, we can define the relevant equations for node
The currents can all be defined via Ohm's LawThis can be done similarly for currents in node All equations can then be expanded and written in terms of voltagesFor node For node
Which then can be turned into Matrix format
Linear solving techniques such as Gaussian Elimination can be used to solve the equationSome nodes can be combined when there is no difference in current or voltage into super nodesThese super nodes can make analysis easierConsider the following circuitNodes and can be defined in terms of Nodes and can be combined into one super node since there is no current difference between them

Node Voltage Analysis Example_0

Sun, 11 Oct 2026 17:41:44 GMT

Circuit Linearity Example_0

Sun, 11 Oct 2026 02:21:27 GMT

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Complex Exponents

Sat, 10 Oct 2026 10:45:21 GMT

For there can be distinct states of complex values for

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Coupled Oscillator Example_3

Thu, 08 Oct 2026 19:52:48 GMT

Drift of Charge Carriers

Thu, 08 Oct 2026 02:35:49 GMT

Drift is the motion of the charge carrier caused by an Electric Field in a Conductor materialIn no electric field, the instantaneous velocity of a carrier can be given by its thermal energyWhile the individual velocity of an electron is random, in a electric field, the average velocity is non-zero and has a preferred directionThe drift momentum , not to be confused with , of the electron can be defined asWhereSo that the velocity of the electron in an electric field can be defined as
The drift velocity can be defined as a function of its Drift Mobility

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Thermal Equilibrium

Thu, 08 Oct 2026 01:27:33 GMT

Thermal equilibrium is the lowest energy configuration in the presence of heat or thermal agitation, Generation and Recombination is only possible due to thermal excitationIn thermal equilibrium, recombination rate is equal to the generation rate
Because a large concentration of electrons will be dealt, statistical treatment is required for Electron or Hole populations

Ionization Energy

Thu, 08 Oct 2026 01:22:45 GMT

Ionization energy is the energy it takes to remove an electron from an atom or ionIonization energy tends to increases across a period and decreases down a groupThe first ionization energy is always lower than the second , since removing an electron from a cation requires more energy

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E-K Diagrams

Thu, 08 Oct 2026 01:03:31 GMT

For a single free electron, the relationship between the kinetic energy and the wave vector is given by an E-k DiagramThe faster the the particle is , the more energy the particle hasThe Diagram is given below
The wave of a particle with respect to can be defined using Bloch's TheoremSo that the probability of detecting an electron at a specific point in a crystal is a periodic function of Since the lattice constant equally displaces atoms, the probability for positions displaced by are equally likely
For any given only certain energy levels are accessible to an electron according to the Quantum Theory

Mass in Particles

Thu, 08 Oct 2026 01:00:10 GMT

The mass of a particle describes how that particle resists changeThe electrons and holes in a crystal interact with a periodic Columbic Field in the crystal, this is due to Bloch's TheoremSince the masses and surf over the periodic potential of the crystal these masses are not the same as the mass of the particle in free space
The effective mass of an electron in a band with a given E-K Relationship isNegative mass Since electrons behave as particles, they are capable of having negative mass Negative charges with negative mass move in the same direction as positive charges with positive mass For a band centered at

Electron-Hole Pair

Wed, 07 Oct 2026 21:56:34 GMT

As the temperature of the semiconductor raises from electrons can raise from the valance band to the conduction band, the empty state left in the valance band is called a holeThe pair of the electron in the conduction band and the hole in the valance band is called an electron-hole pair

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Thevenin and Norton Theorems Example_3

Wed, 07 Oct 2026 16:40:29 GMT

Thevenin and Norton Theorems Example_2

Wed, 07 Oct 2026 16:25:26 GMT

Thevenin and Norton Theorems Example_0

Wed, 07 Oct 2026 16:17:03 GMT

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Thevenin and Norton Theorems Example_1

Wed, 07 Oct 2026 16:14:58 GMT

Source Transformation

Wed, 07 Oct 2026 16:10:23 GMT

Consider a voltage source
According to Kirchhoff's Voltage LawSo that it is equivalent to the current source

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Thevenin and Norton Theorems Example

Wed, 07 Oct 2026 16:03:20 GMT

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Fourier Signal

Tue, 06 Oct 2026 23:40:17 GMT

A Fourier signals exists ifAnd as the Fourier coefficients and approach infinite and the error signal energy approach zeroA Fourier signals exists if has a finite number of of maximums and minimums in one period has only a finite number of discontinuities in one periodThen there exists a Fourier signals whereFor all where is continuous

Coupled Oscillator Example_2

Tue, 06 Oct 2026 19:37:53 GMT

Coupled Oscillator Example_1

Tue, 06 Oct 2026 19:37:22 GMT

Exam 1 Review Example

Tue, 06 Oct 2026 19:01:41 GMT

Diode Example_0

Tue, 06 Oct 2026 16:29:40 GMT

EquilibriumForward BiasReverse Bias

Diode Example

Tue, 06 Oct 2026 16:22:00 GMT

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P-N Junctions Example_1

Tue, 06 Oct 2026 16:17:06 GMT

P-N Junctions Example_3

Tue, 06 Oct 2026 15:51:46 GMT

Crystal Lattice

Mon, 05 Oct 2026 20:09:22 GMT

A crystal lattice is a repetitive pattern that appears throughout the entire crystal in a 3D arrangementUnit cells are the building block of a Crystal LatticeA unit cell with lattice points at its corners and is the smallest possible cell within a latticeA unit cell that does not necessarily have lattice points at its cornersBravais Lattices are infinitely repetitive arrangements of discrete points that can be repeated via privative translations in Vector Space In 3D, there are 14 Bravais Lattices In 2D, there are 5 Bravais LatticesCubic Lattices are lattices where the unit cell is a cubeThe parameter defines the length of one side of the cube and is measured in Angstrom

The diamond lattice, also known as a zinc blende structure, is common for many Semiconductors it is composed of two atoms on alternating sites. Another way of visualizing the diamond structure is a fcc structure with an extra atom placed at from each of the fcc atoms

Circuit Linearity Example_4

Mon, 05 Oct 2026 16:22:13 GMT

Circuit Linearity Example_3

Mon, 05 Oct 2026 16:07:59 GMT

Circuit Linearity Example_2

Mon, 05 Oct 2026 15:56:57 GMT

Circuit Linearity Example

Mon, 05 Oct 2026 15:51:48 GMT

Circuit Linearity Example_1

Mon, 05 Oct 2026 15:49:27 GMT

Band Diagram Example

Mon, 05 Oct 2026 15:44:50 GMT

Energy

Band Diagram Example_0

Mon, 05 Oct 2026 15:44:20 GMT

DistanceEnergy

Accelerating Frames Example

Mon, 05 Oct 2026 15:43:18 GMT

Alternating Current Example

Mon, 05 Oct 2026 15:42:14 GMT

AC Steady State Example

Mon, 05 Oct 2026 15:42:12 GMT

AC Steady State Example_0

Mon, 05 Oct 2026 15:42:11 GMT

Crystal Lattice Example

Mon, 05 Oct 2026 15:25:05 GMT

Simple CubicBody Centered CubicFace Centered Cubic

Semiconductor

Mon, 05 Oct 2026 15:17:40 GMT

Semiconductors are materials with a electrical conductivity between a conductor and an insulator.
They can be single Element like or Or they can be a compound like
Semiconductors are common around Group I through Group VIDifferent materials have different properties that are application specific Materials such as Silicon and Gallium are prominent across all usesIncreased Usage of Complexity Over the past decades, more and more elements have been used in semiconductor fabrication
Semiconductors are often solids and crystalline in nature and are arranged in a Crystal Lattice

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Damped Harmonic Oscillator Example

Sat, 03 Oct 2026 23:15:14 GMT

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Fourier Series of Trigonometric Functions Example

Fri, 02 Oct 2026 02:25:56 GMT

Complete Signal

Thu, 01 Oct 2026 23:50:24 GMT

When the set is such that the error signal energy approaches as approaches , for every member of some class of signal , we can say that the set is complete on for that class of SignalsFor a complete signal For

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Orthonormal Approximation Example

Thu, 01 Oct 2026 23:18:31 GMT

Coupled Oscillator Example

Thu, 01 Oct 2026 19:26:21 GMT

Coupled Oscillator Example_0

Thu, 01 Oct 2026 19:26:10 GMT

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Linear System

Thu, 01 Oct 2026 17:21:28 GMT

A linear system follows the principle of SuperpositionFor all , , , and To check if a system is linear, consider two different inputs and and an output If

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P-N Junctions Example_2

Thu, 01 Oct 2026 16:54:38 GMT

P-N Junctions Example_0

Thu, 01 Oct 2026 16:35:37 GMT

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P-N Junctions Example

Thu, 01 Oct 2026 16:24:53 GMT

Photo Conductivity

Thu, 01 Oct 2026 16:18:55 GMT

Photoconductivity is the change in conductivity, and reciprocally, resistance of a material under excess carrier from optical energyIgnoring Minority Carriers While it is acceptable to ignore minority carrier in Equilibrium, under optical excitation, the changes in Carrier Concentrations may be significant enough that it is not advisable to ignore minority carriers

Fermi Level Example

Thu, 01 Oct 2026 16:01:07 GMT

Steady State Carrier Generation

Thu, 01 Oct 2026 15:56:25 GMT

In a Thermal Equilibrium, the rate of generation is the same as the rate of recombination
In Steady State all time derivatives is zeroIf a steady light is shone on the sample, the optical generate rate (assuming no trapping) can be given as
Consider that a is doped with donors, having Electron-Hole Pairs created uniformly at AssumeFind the recombination coefficient
First we can find Equilibrium concentrationConsider the equation of generation with optical generationBy rearranging

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Compensation

Thu, 01 Oct 2026 02:33:16 GMT

A Semiconductor material is assumed as n-type or p-type. At moderate temperatures.
However semiconductors can be Doped with both donors and acceptors, this is called compensationThis technique can be used to make complex devices where a section of a material can be doped with the opposite type to create sections of opposite type semiconductor
If the same number of donors or acceptors are added as the previously doped amount, the material becomes Intrinsic again

Mesh Current Analysis Example_2

Wed, 30 Sep 2026 16:24:35 GMT

Mesh Current Analysis Example_1

Wed, 30 Sep 2026 16:24:23 GMT

Mesh Current Analysis

Wed, 30 Sep 2026 16:18:42 GMT

Mesh current analysis can be used to solve planar circuits using elementary Current meshes
Each loop of current can be defined so that the current of every Element can be a sum of all current loops Consider the circuitConsider the circuit

Mesh Current Analysis Example_0

Wed, 30 Sep 2026 16:06:18 GMT

Mesh Current Analysis Example

Wed, 30 Sep 2026 16:01:27 GMT

Node Voltage Analysis Example_1

Wed, 30 Sep 2026 15:42:46 GMT

Linear Algebra Review

Wed, 30 Sep 2026 00:04:43 GMT

For dimensional Vector SpaceThe inner product, or the dot product, of two vectors can be given asSince the inner product is a dot product, the two vectors are orthogonal if their dot inner product equals zeroThe norm of a vector can be given asTherefore the vector can be normalized by dividing by its normTwo vectors are orthonormal if their normalized vectors equal zero

Linear Time-Invariant Systems

Tue, 29 Sep 2026 23:52:55 GMT

A linear time-invariant, or LTI, System produces output Signals from inputs signals in such a way that the system remains linear and time-invariant
Example of linear time-invariant systems are Linear CircuitsIn general in order for the system to be usefulA zero input response is the output of the system when This can be thought of as the results of the initial conditions of the systemA zero state response is the output of the system when all initial conditions are zeroThe zero state response can also be determined by using the impulse response The total response of the system is the sum of the zero input and zero state responsesWhereas the LTI function can be defined asAssuming that the input starts at the time All conditions immediately before can be defined asAll conditions immediately after can be defined asAt exactlyThe conditions of is continuous and can be defined asThe conditions of is not continuous and can be defined asTherefore the total output at can be defined asGiven the scenarioCompute the valuesFirst define the convolutionAnd define the derivativeBy plugging in we can see thatBy using the initial conditions
Recall the property of impulse signals And that the integral of a function over a finite period of time gives a sum of areas over a infinitely small deltaAssuming a form of We can convert the form into a summation
This form can be turned into a Linear SystemOr a time-invariant system

Steady State Injection

Tue, 29 Sep 2026 16:49:50 GMT

Assume that excess are inject into a infinite Semiconductor bat at
These protons will be diffused throughout the bar recombining with an electron with in time
The rate of Steady State diffusion can be modeled by the formWhere the concentration of can be modeled byAnd and are boundariesSo that the concentration can be simplified toY axisX axis00-10-10-5-5551010-5-555Expression 1As the protons diffuse farther away the chances that the proton is recombined is increases and the overall concentration decreases

Types of Recombination

Tue, 29 Sep 2026 16:43:12 GMT

Electron-Hole Pair recombination can happen through multiple processes, theses are often counterpart to Types of GenerationIndirect recombination happen in multiple stepsRecombination can happen in the R-G center, often called traps, between bands

Types of Generation

Tue, 29 Sep 2026 16:42:54 GMT

Electron-Hole Pair generation can happen through multiple processes, theses are often counterpart to Types of Recombination

Types of Recombination Example

Tue, 29 Sep 2026 16:40:33 GMT

Semiconductors under Illumination

Tue, 29 Sep 2026 16:35:53 GMT

When there is excess photons upon a Semiconductor, excess carriers are createdFor a beam of photons upon a sample of thickness
Assuming that per the Photoelectric EffectThe intensity of light at distance can be foundThe coefficient of absorption will vary with wavelength and semiconductor materialFor longer wavelengths, the absorption is negligible For shorter wavelengths, the absorption is considerable
Since the absorption wavelengths depends on the Band Gap of the material, the wavelength that common semiconductor materials can be foundFor For

Diffusion with Recombination

Tue, 29 Sep 2026 16:10:13 GMT

During, Recombination, Carrier Concentrations can be changed, effecting diffusion rates

Element

Tue, 29 Sep 2026 16:01:07 GMT

Elements can still be broken down to simpler elements, i.e. Ozone to OxygenElements are defined as a substance that is comprised of atoms with the same atomic numberDifferent structural forms of an element that are in the same phaseCarbon : Diamond GraphiteA substance that contains two or more elementsTwo or more substances, which can be elements or compounds, in a physical combinationHomogenous: Substances are dissolved and can not be separated easily Heterogenous: Substances can be separated easily Elements consists of atoms Fusion and ionization can divide atoms All atoms of an given element are identical and mass and other properties Isotopes have different masses and properties Different elements have different masses and properties Compounds are formed when unlike elements are combined in a simple integer ratio (Law of Definite proportions) In a pure compound, the constituent elements are always present in a definite proportionThe ratio of the masses in two different compound are usually simple integersEqual volumes of different gases at the same temperature and pressure contain equal number of particlesCharge of a proton = - Charge of an electronMass of a proton ~= Mass of a neutron Mass of a proton = 1836 * Mass of an electron

Built-In Fields

Tue, 29 Sep 2026 15:58:38 GMT

When a Semiconductor is in Equilibrium, Electron-Hole concentration gradients induces built-in fieldsThese fields act as if they are externalSteady Carrier Concentrations Without a concentration gradient, there can not be built-in fields.

Linear Dependent Source Example_0

Mon, 28 Sep 2026 20:15:16 GMT

Linear Dependent Source Example

Mon, 28 Sep 2026 20:08:15 GMT

Node Voltage Analysis Example

Mon, 28 Sep 2026 16:36:27 GMT

Electrical Model Analysis

Mon, 28 Sep 2026 16:26:49 GMT

Consider the Circuit Model below
Each node can be thought of as an element in a Spanning Tree
Valid Solutions Even if the number of equations Kirchhoff's Equations matches the number of unknown variables, a Unique Solution is not always guaranteed

Tellegen's Theorem

Mon, 28 Sep 2026 16:26:33 GMT

Define and using all passive or all active Wye ConnectionsConsider the circuit below

To solve the circuit, we can generate all Kirchhoff's Equations
By plugging in given and trivial equalitiesThe resulting linear equation can be solved easily
See Gaussian Elimination

Op Amps Example_2

Mon, 28 Sep 2026 16:25:42 GMT

Op Amps

Mon, 28 Sep 2026 16:21:12 GMT

Operational Amplifiers take in a positive and negative voltage and then outputs the difference of that voltage.When analyzing op amp, these general rules can be followedConsider the digital to audio converter with three bits represented in binaryTo represent this, a simple summing amplifier can be usedWith the equation ofA switch will then be added controlled by the bit
So thatThis can be generalized over all bitsNow consider the opposite scenario, where a voltage needs to be turned into digital bits Now consider the property of the op amp that when, provided with a reference voltage, a slight difference between the input voltage and the reference voltage will saturate the op amp in that direction

Op Amps Example_1

Mon, 28 Sep 2026 16:10:48 GMT

Op Amps Example_0

Mon, 28 Sep 2026 16:05:46 GMT

Op Amps Example

Mon, 28 Sep 2026 16:05:13 GMT

Simple Harmonic Motion Example

Mon, 28 Sep 2026 03:13:00 GMT

Simple Resistive Circuits

Sun, 27 Sep 2026 22:41:21 GMT

In simple resistive circuits, the Resistors are linear, so that the Current though them is proportional to the voltage across it.The voltage of an ideal resistor can be given by
Circuits can be generally solved through Kirchhoff's Laws and the voltage constraint defined above. However for a more efficient approach, these measures can be taken
Avoid defining voltages and currents for all Elements, rather just the ones needed for the task Identify parts of the circuits that follow KVL and KCL and other Ohm's equations Solve the resulting equations

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Convolution Example_7

Fri, 25 Sep 2026 00:06:05 GMT

Convolution Example_4

Thu, 24 Sep 2026 23:34:16 GMT

Convolution Example_5

Thu, 24 Sep 2026 23:34:04 GMT

Convolution Example_6

Thu, 24 Sep 2026 23:33:48 GMT

Convolution Example_3

Thu, 24 Sep 2026 23:30:46 GMT

Carrier Concentrations Example

Thu, 24 Sep 2026 17:18:03 GMT

Fermi-Dirac Statistics

Thu, 24 Sep 2026 00:00:19 GMT

Fermi-Dirac Statistics is a way to determine the probability of finding Electrons or Holes within a SemiconductorProbability of an ElectronProbability of a hole

Ionization Energy

Wed, 23 Sep 2026 23:53:10 GMT

Ionization Energy is the Energy required to ionize a donor atom, or to free the extra electron to leave a positive ion
Within a Band Diagram, ionization energies above the Fermi Level are N-Type or donors and ionization energies belove the Fermi Level are P-Type or acceptors

Convolution Example_2

Wed, 23 Sep 2026 00:12:38 GMT

Convolution Example_1

Wed, 23 Sep 2026 00:05:07 GMT

Convolution Example_0

Tue, 22 Sep 2026 23:18:26 GMT

Convolution Example

Tue, 22 Sep 2026 23:15:45 GMT

Forced Harmonic Motion Example

Tue, 22 Sep 2026 19:36:49 GMT

Simple Harmonic Motion Example_5

Tue, 22 Sep 2026 19:26:45 GMT

Damped Harmonic Oscillator Example_0

Tue, 22 Sep 2026 19:26:41 GMT

Intrinsic Material

Mon, 21 Sep 2026 20:09:21 GMT

If there are no Dopants present, then the Semiconductor is called intrinsic
Because there are no Free Carriers at and above , Electron-Hole Pairs are generated, we have Free Carrier Generation above The number of electrons and holes in the semiconductor is thermally dependentSo thatThis means that there will always be a ratio for electrons and holes in free carrier generation, change is then not generated during free carrier generation

Free Carrier

Mon, 21 Sep 2026 20:00:10 GMT

A free carrier is an electron which have been broken from the lattice in the process of Free Carrier GenerationAbove the electron will have absorbed enough energy, equal to to have its bonds broken
The electron will leave from the valance band to the conductive band, creating a hole within the valance band, the holes and electrons are generated as pairsThe holes left behind in the balance band will behave as particles

Free Carrier Generation

Mon, 21 Sep 2026 18:00:44 GMT

Breaking the bonds in a crystal requires energyThe energy of breaking a bond is equal to the Band Gap Energy
By investing this energy into the lattice, an electron can be removed from an otherwise satisfied bond state, that removed electron will roam as a Free Carrier, this process is called free carrier generation

Photoelectric Effect

Mon, 21 Sep 2026 17:28:29 GMT

The photoelectric effects happens as electrons are ejected from a metal surfaceClassical theory determined that the energy of light depends only on its intensity and not its frequency, however experimentation showed that electrons are emitted, even at a low intensity, given a high enough frequencyEinstein relates the energy of photons to its frequency rather than its intensity Where is the Planch constant and is the frequency of the photonAtoms emit or absorb light based on energy released or absorbed, this relation can be given by

Mapping Properties

Fri, 18 Sep 2026 00:46:36 GMT

Both the real and imaginary parts of a nonzero Complex Number are equally important for determining its position on the Complex planeBy taking a complex number We can see thatWhereThe length of only relies on the real part of whereas the imaginary part "rotate" the complex vector around the origin

Source Transformation Example

Wed, 16 Sep 2026 16:50:09 GMT

Source Transformation Example_0

Wed, 16 Sep 2026 16:49:43 GMT

Resistor Equivalent

Wed, 16 Sep 2026 16:28:26 GMT

The current can be found viaFrom there the voltage across a specific can be solvedResistors in series can act as voltage dividersConsider the circuit belowThe resistors act as a voltage divider where the output voltage will beVoltage Divider Stable Voltage Recall that the output voltage of a voltage divider is The closer the ratio is The the closer the output voltage to the source voltage Resistors in parallel can act as current dividersConsider the circuit belowThe resistors act as current dividers, where the equation for the current isWhere

Resistor Equivalent Example_0

Wed, 16 Sep 2026 16:25:50 GMT

Resistor Equivalent Example_1

Wed, 16 Sep 2026 16:24:55 GMT

Resistor Equivalent Example

Wed, 16 Sep 2026 15:57:41 GMT

Common Signals

Wed, 16 Sep 2026 00:17:42 GMT

Y axisX axis00-10-10-5-5551010-5-555Expression 1The unit step function can be used as a define domains of signalsConsider the functionThis can be represented with two unit step functionsConsider the piecewise functionThis can be converted to the equivalent functionY axisX axis00-10-10-5-5551010-5-555Expression 1By multiplying a signal by the unit impulse, we can get the signal as a coefficient of the unit impulseBy multiplying a signal by the unit impulse and integrating, we can get the signal value at that timeConsider the functionConsider the functionWhereSo thatAny given signal can be decomposed into its even and odd componentsConsider the decomposed function

Signal Operations

Tue, 15 Sep 2026 23:24:36 GMT

Signals can be operated upon by operations with single input operations to operations with multiple inputsConsider the simple signal Y axisX axis00-10-10-5-5551010-5-555Expression 1A time shift changes the phase of the signalY axisX axis00-10-10-5-5551010-5-555Expression 1A time scale changes the scale of the signalWhere does not changeY axisX axis00-10-10-5-5551010-5-555Expression 1A time reverse flips the time domain of the signalY axisX axis00-10-10-5-5551010-5-555Expression 1Consider the signals and Y axisX axis00-10-10-5-5551010-5-555Expression 1Y axisX axis00-10-10-5-5551010-5-555Expression 1To perform the operationYou can use either of the two methodsTo perform the operation

Simple Harmonic Motion Example_4

Tue, 15 Sep 2026 19:20:32 GMT

Simple Harmonic Motion Example_1

Tue, 15 Sep 2026 19:06:08 GMT

Simple Harmonic Motion Example_3

Tue, 15 Sep 2026 19:05:53 GMT

Simple Harmonic Motion Example_2

Tue, 15 Sep 2026 18:53:57 GMT

Accelerating Frames

Tue, 15 Sep 2026 18:50:24 GMT

In an accelerating frame, is not The rotation change in is equal to And since we can simplifySo that for any vector its inertial Time Derivative is equal toThe position in the inertial frame can be given by The Foucault Pendulum
It has a velocity of But because the earth is spinning on its axis

Direct and Indirect Semiconductor

Tue, 15 Sep 2026 16:54:57 GMT

Direct transitions Within indirect transitions, the transition from and do not occur at the same Trap States In silicon processing traps states can occur when silicon crystals are impure, leading to a higher concentration of indirect transitions

E-k Diagrams Example

Tue, 15 Sep 2026 16:45:53 GMT

Band Gap

Tue, 15 Sep 2026 16:41:33 GMT

A band gap is the area within two bands within a Band Diagram that indicates the amount of energy required for an electron to move between the two bands, it is denoted by
The band gap tends to decrease as the temperature of the system increases to the a higher amplitude of Atomic Vibration due to the excess energy

Energy States and Bands

Tue, 15 Sep 2026 16:37:49 GMT

Electrons in an Atom have discrete energy levels, however electrons in a solid are restricted to range of energies called bandsThis is due to electrons acting as Quantum Particles within a solidIn a system with on atom, the energy levels of the electrons may look similar to the following
In a system with multiple intersecting atoms, the Wave Functions of the electrons will overlap the their potential energies will sumThe system will balance into a stable distance
In silicon, the and states go though Band Splitting when atoms are brought together to generate states
At , lower states are filled with valance electrons and the higher states act as Conductors
Because of this Silicon is able to share 4 of its electrons with its neighbors with 8 overall shared electrons in a Covalent Bond
At the Band Gap of the Silicon is , since the electrons seek the state with the least energy required, all electrons will be in the valance band .At there will be enough energy to jump to the conduction band

Energy States and Bands Example_1

Tue, 15 Sep 2026 16:31:24 GMT

Band Diagram

Tue, 15 Sep 2026 16:24:22 GMT

A band diagram A band diagram can be simplified by taking the highest partially filled state and lowest partially filled stateBand Diagram Example_0.excalidraw

Energy States and Bands Example_0

Tue, 15 Sep 2026 16:10:12 GMT

Energy States and Bands Example

Tue, 15 Sep 2026 16:06:54 GMT

Quantum Mechanics in Solid State

Tue, 15 Sep 2026 16:01:47 GMT

The electrons propagating within a system behaves as particles within a physical systemSchrodinger's Wave Equation can be used to describe the movement of electronsThe Pauli Exclusion Principle dictates that no two electrons in a interacting system can have the same Quantum NumbersP

Metallic Bonding

Tue, 15 Sep 2026 15:50:07 GMT

In metallic bonding, a cloud of delocalized electrons bond with the positive ions of a metal

Bonding in Solids

Tue, 15 Sep 2026 15:48:29 GMT

Solids can bound via Ionic Bonding, Metallic Bonding, or Covalent Bonding

Silicon Electronic Structure

Tue, 15 Sep 2026 15:47:02 GMT

Silicon has 14 electrons organized in a electron structure

Tellegen's Theorem Example_0

Mon, 14 Sep 2026 16:39:06 GMT

Tellegen's Theorem Example

Mon, 14 Sep 2026 16:33:36 GMT

Electrical Model Analysis Example_2

Mon, 14 Sep 2026 16:20:02 GMT

Electrical Model

Mon, 14 Sep 2026 16:15:51 GMT

Electrical Models are used to approximate real world circuits via modeling elements and connections
A Current is the flow of positive charge through a circuit, a positive current indicate electrons flowing in the opposite direction, it is represented by and is measured in Amperes
See SI Units for more detailsCurrent Direction It's always best practice to label the direction of the current to avoid confusion
The Voltage is the potential of positive charge between two terminals, it is represented by and is measured in Volts
See Ohm's Law
Power is the work being done by charge in a circuit, it is represented by and is measured in WattsPower Dissipation and Supply Because power the change in energy it may be positive or negative
A Voltage Source is an element that supplies voltage into a circuit
A Current Source supplies current into a circuit
A Resistor is an element that limits current by dissipating PowerThe Mho The inverse of the Ohm is unofficially called the Mho

Electrical Model Analysis Example_1

Mon, 14 Sep 2026 16:15:02 GMT

Electrical Model Analysis Example_0

Mon, 14 Sep 2026 16:14:03 GMT

Electrical Model Analysis Example

Mon, 14 Sep 2026 16:08:01 GMT

Kirchhoff's Laws

Mon, 14 Sep 2026 16:04:12 GMT

Kirchhoff's Laws dictate how electricity moves within a circuitKirchhoff's Current Law states that the current going into a node or component must be the same as the current going out of the node or componentConsider these two scenariosIn both cases the total current into the system equals the total current out of the system Kirchhoff's Voltage Law states that all voltage rises must be accompanied by voltage dropsAnd therefore the sum of all voltage across a circuit must sum up to zeroConsider this Loop containing three components Since the total rises must equal the total drops , therefore Loops within Loops The equivalent KVL of a bigger loop is the sum of all independent loops within it

Exponential Function

Sun, 13 Sep 2026 19:40:10 GMT

A real-valued function of has the following properties Continuous on given as Now consider a exponential complex-valued function

Power and Energy in Signals

Sat, 12 Sep 2026 17:03:04 GMT

Power is the the rate of supplying or absorbing energy often measured in in Watts
By applying Chain RuleWe get the term which is the formal definition of the volt
And the term which is the definition of CurrentSo that the power signal can be defined asBy integrating power we can find the overall energy of the signal

Signal Classifications

Sat, 12 Sep 2026 16:53:34 GMT

Signals can be classified into a number of groupsA signal with continuous time can take on any valueA signal with discrete time can only take quantified valuesAn analog signal can take on any valueA digital signal can only take discrete valuesPeriodic signals are repetitive indefinitelyFor some The smallest value of is the period of that signal A periodic signal can be generated by periodic extension of a small segment of that signal
Periodic signals are always NoncausalAperiodic signals do not repeatA signal is causal if for all A signal is not causal if for some A signal is anti-causal if for all
A signal is an energy signal if is finite
A signal is power signal if if finite and greater than Energy Power Inverse Because power and energy signals are inverses One signal cannot satisfy the definition for both, it must either be a power or an energy signal Likewise, a signal cannot dissatisfy the definition for both A deterministic signal has a given value for each given key
A random signal is not known exactly but known Probabilistically and information bearing

Energy of a Signal

Thu, 10 Sep 2026 23:50:42 GMT

Signal

Thu, 10 Sep 2026 23:49:06 GMT

A signal can be defined as a function of independent variablesData signals can be thought of as functions of time and spaceA system is a process that intakes a signal and outputs a signalAn amplifier can take a signal and amplify it While a television can take a signal and change its form
The Energy of a signal can be given as
Where as for Complex signals it is given asOn the condition thatConsider the signal
The energy of the signal in the domain can be foundThe power of the signal can be determined via the mean squared value of
Where is the Root Mean Square of the signalConsider a periodic saw signal
The period is Then the power can be foundThe root mean squared is then

Signal Classifications Example_1

Thu, 10 Sep 2026 23:42:04 GMT

Signal Classifications Example_2

Thu, 10 Sep 2026 23:41:47 GMT

Signal Classifications Example_0

Thu, 10 Sep 2026 23:40:38 GMT

Signal Classifications Example

Thu, 10 Sep 2026 23:39:25 GMT

Resistor

Thu, 10 Sep 2026 23:36:26 GMT

A resistor adds resistance to a line of current, any medium that can resist a current (like a wire) can be treated like a resistor, resistors always absorb energy Resistors are often the most ideal componentsideal: Ideal components do not draw current nor supply current A resistor is given by this symbolEurope Standards In Europe, resistors are defined by a different symbol The resistance of a resistor can be determined by using a color code system.Ohm's Law relates voltage and current through a resistor Ohm's Law Ohm's Law is only meant to be used for resistors, therefore it can not be generalized to non resisting components The root mean square gives the mean of a set of, of continuous, numbers This is useful for measuring sine/alternating waves in AC voltagesResistors in series combine their resistanceResistors in parallel divide the current between all resistorsJustification for resistors in parallelIf then If then For resistors in series, the total resistance is always greater than the largest value For resistors in parallel the total resistance is always lesser than the smallest valueIn certain cases, the resistance of an entire circuit (in series) can be combined into a single resistor
The resistance of a resistors can be variable by shifting the contact between two points, changing the resistance
An electric guitar uses variable resistors to control volume and tone
The temperature coefficient of a resistor is the variation in resistance given changes in temperature

An ohmeter measures the resistance by measuring the voltage of a predictable current source

Signal Example_0

Thu, 10 Sep 2026 23:32:29 GMT

Signal Example

Thu, 10 Sep 2026 23:23:26 GMT

Simple Harmonic Motion Example_0

Thu, 10 Sep 2026 19:04:26 GMT

Wafer Orientation

Thu, 10 Sep 2026 16:49:53 GMT

Semiconductor wafers can be orientated in predefined crystallographic plane based on the desired propertiesBecause of the differences in electron structure, the mechanical property of the silicon changes based on orientation
Typical Directions A notch or a flap on the side of the wafer can dictate the orientation of the wafer. Wafers are commonly (notated in Miller Indices)

Miller Indices

Thu, 10 Sep 2026 16:49:52 GMT

Miler Indices are three integers that defines a crystalTo find Miller IndicesFind the intercepts of the plant with the crystal axis'Then take the reciprocal of the three integers and reduce to the smallest set of integers

Crystal Plane

Thu, 10 Sep 2026 16:44:29 GMT

Hard Sphere Approximation

Thu, 10 Sep 2026 16:33:50 GMT

We can approximate Cubic Lattices as hard spheres packed into a unit cellEach atom has a radius ofEvery Unit Cell contains 8 corner atoms, and 6 face atomsThe packing factor is a ratio of the volume of atoms within the cell and the volume of the cell

Semiconductor Example

Thu, 10 Sep 2026 16:05:02 GMT

Vector Space

Thu, 10 Sep 2026 15:43:44 GMT

Vector Space is

Ohm's Law

Wed, 09 Sep 2026 16:32:32 GMT

Ohm's Law describes the relationship between a voltage difference between two points in a Conductor and the current and resistance through which the electrons flow between the two points

Electrical Model Example_0

Wed, 09 Sep 2026 16:25:47 GMT

Electrical Model Example

Wed, 09 Sep 2026 16:18:50 GMT

Schrödinger's Equation

Wed, 09 Sep 2026 15:49:16 GMT

Because electrons are waves, it's position and movement in space can be represented by a wave equationThe Schrödinger's equation is underivable and is regarded as a postulate.Its general form can be given byThe solutions to this equation are the wavefunction and energy of a particle
The Probability Density of a particle can be given by The solution for a hydrogen and other one election ions gives the wavefunction and corresponding energiesThe wave functions are called atomic orbitals to differentiate them from Bohr orbitsWhere is the atomic number And is the Rydberg energy ()
Although the energies are the same as levels predicted by Bohr's theory, quantization is a natural consequence of the Schrödinger equation

Impedance

Wed, 09 Sep 2026 15:48:45 GMT

Impedance is a generalization of resistance in AC circuits, it is generally represented byWhere is a sinusoidal voltage And is a sinusoidal currentSimilar to resistance, impedance is measured in
A Resistor will always have an impedance of its resistanceFor dynamic elements such as capacitors and inductors, their resistance can be calculated via using Let's consider the voltage of both components
We assume that the current is always a complex sinusoidal function Simplifying Per the form

Complex Number Calculation

Wed, 09 Sep 2026 00:14:12 GMT

Certain forms of Complex Numbers are more convenient for certain calculationsFor example for addition, the Cartesian Form is more convenientFor multiplication, the Polar Form is more convenient
For division, the Complex Conjugate can be usedFor powers and roots, the Polar Form is more convenientFor larger values of the equation might require multiple answers

Wave Function

Wed, 09 Sep 2026 00:02:24 GMT

The wave function of a single Qubit system can be defined asIn the function, and are complex numbers Where is the probability that the qubit is a and is the probability that the qubit is a Consider the wave functionThe function has exactly a chance of being a and chance of being a The form of a wave functionCan be expanded to the form
Where , , and are all real numbersThe coefficient has no observable effect on the spin of the qubit and thus can be ignored Giving the complex equation

Complex Number Example_2

Wed, 09 Sep 2026 00:01:11 GMT

Complex Number Example_0

Wed, 09 Sep 2026 00:01:06 GMT

Complex Number Example

Wed, 09 Sep 2026 00:00:56 GMT

Complex Number Example_1

Tue, 08 Sep 2026 23:45:34 GMT

Time Domain Analysis

Tue, 08 Sep 2026 23:25:31 GMT

Required Textbooks

Tue, 08 Sep 2026 22:52:38 GMT

Required: B. G. Streetman, S. K. Banerjee, Solid State Electronic Devices, 7th ed., Pearson/Prentice Hall, 2014. ISBN-10: 0133356035. ISBN-13: 978-0133356038.Required (free online): C. C. Hu, Modern Semiconductor Devices for Integrated Circuits, 3rd ed., Wiley, 2007. (https://www.chu.berkeley.edu/modern-semiconductor-devices-for-integrated- circuits-chenming-calvin-hu-2010/)Required: George King’s Vibrations and Waves (Units 1-2). The quantum waves portion of the course (Unit 3) will be taught from other resources that will be provided at a later date. Recommended: French’s Vibrations and Waves (Units 1-2).

Electrochemical Series

Tue, 08 Sep 2026 20:48:24 GMT

Substrates that is in contact with solvents has a specific Solution Pressure. An oxidation reaction will happen until the ions exerted on the substrate matches the solution pressure
The Nernst Equation defines how the a potential of a reaction is affected by the changes in concentration of the reaction partner. This can be used to calculate the electromotive force of a Galvanic CellWhere is the electrode potential in is the standard electrode potential
is the Gas Constant is the temperature in is the charge number (number of transmitted electrons)
is Faraday's Constant is Activity ( oxidized form, reduced form)

Galvanic Cell

Wed, 02 Sep 2026 04:14:28 GMT

A Galvanic Cell, also known as voltaic cells, produce energy by spontaneous oxidation-reduction.Galvanic Cells are organized into three groups Primary Secondary Fuel Consider a solution silver nitrate in which a bar copper of is added, the ensuing reaction is a spontaneous oxidation and reduction that produces energy which is drawn as current through the coperOxidation Half-Equation
Reduction Half-EquationProviding the full reactionNote that the reduction half equation was doubled to balance the equation

Transistor

Mon, 24 Aug 2026 00:12:46 GMT

A transistor is a semiconductor device used to switch electronic signals.

Thyristor

Mon, 24 Aug 2026 00:06:04 GMT

A thyristor is a similar to a Transistor but with bistable behavior. It consists of four differentially doped semiconductor layers that creates three Depletion Regionsbistable: two stable statesThe outer regions connect to the anode and cathode terminals while the inner region connects to the gate terminalWhen connected to a gate voltage, the thyristor fire and behaves like a diode in the forward direction. If the current flowing between the anode and cathode drops below the holding current, then the thyristor returns to a blocking state.A thyristor is given by this symbol.

Thyristor Example

Thu, 06 Aug 2026 20:10:22 GMT

Hall Effect Sensor

Thu, 06 Aug 2026 20:01:37 GMT

The hall effect sensor creates voltage via a Current and a Magnetic FieldThe generated voltage is proportional to the strength of the magnetic field
When there is a present magnetic field, positrons are routed forwards the positive voltage side

Hall Effect Sensor Example

Thu, 06 Aug 2026 20:01:22 GMT

Diode

Wed, 05 Aug 2026 17:48:54 GMT

A diode is an active component that only allows current through one way of the componentDiodes with a current above 1 Amp are referred to as power diodes Some diodes emits light or Electromagnetic Waves, the color or frequency of the wave depends on the semiconductor material and the doping of the material.Zener diodes have a sharp bend in the reverse voltage side that allows reverse current.

Equations

Sat, 01 Aug 2026 21:48:20 GMT

Current Law: Voltage Law: Series: Parallel: Series: Parallel: Voltage Divider: Current Divider: RC circuit: RL circuit: Natural Response (1st order): Resonant Frequency: Ideal assumptions: (virtual short), (no input current)Diode (Shockley): where at room tempBJT: MOSFET (Saturation): where MOSFET (Triode/Linear): De Morgan's: Key Identities: Euler's: → Multiply: magnitudes multiply, angles add. Divide: magnitudes divide, angles subtract.Transfer Function:

Integrated Circuits

Sat, 01 Aug 2026 21:22:48 GMT

Sources

Sat, 01 Aug 2026 21:22:02 GMT

Sources are Nodes that provide a voltage or Current for the circuitAn ideal voltage source maintains a constant voltage no mater the change in circuit parametersIn a circuit, the voltage of the circuit is always equal to the voltage of the source
In reality a voltage source can not generate power nor provide infinite energyFor a short circuit, the circuit voltage is zeroFor an open circuit, the current is zeroThe power supplied to the load by a real voltage source is as followsMaximum Power The maximum power can be transmitted if the load resistance is equal to the internal resistance of the source An ideal current source maintain a constant current throughout the circuit
A real current source can not generate power nor provide infinite energyFor a short circuit, the voltage is zeroFor an open circuit, the current is zero

Sources Example_0

Sat, 01 Aug 2026 21:14:20 GMT

Sources Example

Sat, 01 Aug 2026 21:02:19 GMT

Node

Sat, 01 Aug 2026 20:57:18 GMT

A node is a junction with several paths, nodes are subject to Kirchhoff's Laws
Nodes are area of wire that connect two or more components, uses KCL
Super Nodes are multiple nodes linked together (often linked with an ideal voltage source), uses KCL
Loops can be thought of as an independent system of voltages, all voltage additions must be accompanied by voltage drops, uses KVL

Wye Connection Example

Sat, 01 Aug 2026 20:53:21 GMT

Wye Connection

Sat, 01 Aug 2026 20:51:00 GMT

Wye and Delta connections are often used in AC circuits

Rotational Energy

Wed, 29 Jul 2026 21:31:13 GMT

Rotational kinetic energy is the energy stored in a rotating object, similar kinetic energy stored in a moving objectWhere is the moment of inertia and is the angular velocityFor a point mass at distance from the axis of rotation:

General Roation in 3D

Wed, 29 Jul 2026 21:30:59 GMT

Force can be defined as a sum of radial and tangential forcesSince Because rotation induces an outward radial force on its subject, there must be a counteractive force to ensure the subject stays at the same radial positionAnd since Simplifying in polarBecause is constant, substitute the variable to constants

Rehashing

Wed, 29 Jul 2026 21:28:35 GMT

Rehashing is the process of resizing a Hash Table when it becomes too fullWhen the load factor exceeds a threshold, a new table with approximately double the capacity is allocated and all existing entries are re-inserted using a new hash function
Rehashing takes time so must be balanced to ensure that rehashing is done as few times as possible while maintaining low collision rates

Hash Table

Wed, 29 Jul 2026 21:27:54 GMT

A table that stores objects in hashed table indexesDefine a table (Array) of length Define a Hash Function that maps keys into an integer indexpublic static int hash( String key, int tableSize ) { int hashVal = 0; for( int i = key.length()-1; i >= 0; i-- ) hashVal = 37 * hashVal + key.charAt( i ); hashVal %= tableSize; if( hashVal < 0 ) hashVal += tableSize; return hashVal; Several parameters can be hashed by multiplying the hash function of each parameter by a unique prime numberInputs of Hash Functions While any object can be hashed, the object but be immutable in order to ensure the same object can retrieve the same index
If several objects return the same hashed index, the object is then prepended to a Linked List at that indexIdeally the the length of the linked list at every index will be 1, so the search function will be
In the worst case scenario, the hashing function will return the same index for all objects, the search function will be The average length of the linked lists is given by
In open hashing, each table cell holds a container (Linked List) and all items that hash to that cell are stored in that containerOpen Hashing provides better insertion and search speed at the cost of removal speed and memory cost `
In closed hashing, all items are stored directly in the table, and a Probing function is used to determine the exact cellClosed Hashing provides better deletion speed and lower memory cost

Divergence Theorem

Wed, 29 Jul 2026 21:23:22 GMT

The divergence theorem relates the flux of a vector field through a closed surface to the volume integral of the divergence of the field insideWhere is a closed region in , is its piecewise-smooth boundary oriented outward, and is a continuously differentiable vector field

Induction

Wed, 29 Jul 2026 21:21:54 GMT

Mathematical induction is a proof technique used to prove statements about all natural numbers. It consists of a base case and an inductive step, where assuming the statement holds for proves it holds for Assumes that any induction is true for all integersAny integer has factorization into prime numbers is a prime number can be factored into itself is not a prime number can be written as a since and can be factored into primes, can be factored into primesTo prove that proof is true, assume is falseThere are infinite prime numbersAssume there are finite prime numbers. Prime numbers are Define integer One of the primes must but a factor of N (all integers can be factored into primes) There must exist an integer where So that divides into Since divides and product = 1 There can not be finite prime numbers

Tree

Wed, 29 Jul 2026 21:20:56 GMT

Trees are data structures that are ordered in a hierarchical or nested structureRoot: A node at the top of the tree Subtree: A smaller tree under a root tree Leaves: Nodes at the end of branchesParent: Nodes that contain one or many sub nodes Child: Nodes that is contained in a parent node Sibling: Nodes that share a parents Ancestors: All nodes that contain the current node including itselfPath: The sequence of nodes where the two nodes are connected via the edges Length: The number of edges on the pathDepth: The length of the path from the root to the node Height: The length of the longest path from the root to a leafDiameter: The number of nodes on the longest path between any two nodes Maximum Width: The number of nodes on the level in the tree that contains the most nodesData can be represented in trees to speed up algorithms, because it's organized only parts of a tree need to be considered to solve problems Tree can often speed up an algorithm from to An infix expression can be represented with a tree, for example the equation can be represented byTrees can be stored in a contiguous array using an address scheme where each node is assigned an indexFor a complete binary tree stored in an array starting at index 1:This scheme allows navigation between nodes, different trees will use different index mappings depending on its needsA tree can be represented by a Linked List by referencing a sibling and the first childpublic class LinkedSiblingTree { private TreeNode root; private static class TreeNode { E element; TreeNode firstChild; TreeNode nextSibling; } } Any tree can be traversed in different waysIn level order, nodes are traversed from left to right, top to bottom

Gradient

Wed, 29 Jul 2026 21:10:22 GMT

Chain rule of can be defined as a dot product of a Gradient and a derivative
Let be completely contained in the Level Set Then is always
Since the gradient is orthogonal to every tangent vector in the Level SetFind an equation for the tangent plane to the surface at the point The magnitude of the gradient is the maximum rate of change in the direction of Only maximized when or whenThe del operator is a vector differential operator that generalizes the notion of a derivative
Applied to a scalar field , the del operator produces the Gradient
The del operator can also be applied via dot product to produce divergence and via cross product to produce curl

Initial Value Problem

Wed, 29 Jul 2026 21:09:15 GMT

Initial value problems provide a unique solution to a differential equation when an initial condition is specified.

Compiler

Wed, 29 Jul 2026 20:54:48 GMT

A compiler is a program that turns code into binary code, complied programs are often faster and more portable than codeIn windows cmake can be used to compile C files In linux, gcc can be used Source file#include int main() { printf("%s\n", "Hello, world!"); return 0; } Expands #include and #define header Replaces all static variables with their values Headers The complier only adds the implementation of functions within a header when they are utilized within the code When using standard libraries like stdio and stdlib, only the used functions within the library are included to reduce compiled size Preprocessed Fileextern int ftrylockfile (FILE *__stream) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))); extern void funlockfile (FILE *__stream) __attribute__ ((__nothrow__ , __leaf__)) __attribute__ ((__nonnull__ (1))); # 949 "/usr/include/stdio.h" 3 4 extern int __uflow (FILE *); extern int __overflow (FILE *, int); # 973 "/usr/include/stdio.h" 3 4 # 5 "helloworld.c" 2 # 6 "helloworld.c" int main() { printf("%s\n", "Hello, world!"); return 0; } Translates preprocessed into readable assembly code Complied File.file "helloworld.c" .text .section .rodata .LC0: .string "Hello, world!" .text .globl main .type main, @function main: .LFB0: .cfi_startproc endbr64 pushq %rbp .cfi_def_cfa_offset 16 ... Translate the code into machine code Assembled File�UH��H�H��]�Hello, world!GCC: (Ubuntu 14.2.0-4ubuntu2) 14.2.0GNU �zRx E �U ��helloworld.cmainputs �� .symtab.strtab.shstrtab.rela.text.data.bss.rodata.comment.note.GNU-stack.not e.gnu.property.rela.eh_frame @ �0 ▒&^^1^90l&B �R�j�e@�▒ ▒�� ▒ �▒� Links object files and outputs the final file Overloading In C, the linker associates each symbol as a single definition, thus functions can not be overloaded Linked File@ @@@�▒▒▒��uu ��-�=�=X`�-�=�=�8880hhh��S�td8880P�td 44Q�tdR�td�- �=�=HH/lib64/ld-linux-x86-64.so.2 GNU��GNU��d��'�+��U��`~▒��FDO{"type":"deb","os":"ubuntu","name":"glibc ","version":"2.40-1ubuntu3","architecture":"amd64"}GNU��e�mH d s ▒"puts__libc_start_main__cxa_finalizelibc.so.6GLIBC_2.2.5GLIBC_2.34_ITM_deregisterTMCl oneTable__gmon_start___ITM_registerTMCloneTable'u▒i 1��=�@�?�?�?�?�?�? ��H�H��/H��t��H��5�/�%�/@��h��f��% �/fD��%v/fD��1�I��^H��H��PTE1�1�H�=��S/�f.�H�=y/H�r/H9�tH�6/H��t ��H�=I/H�5B/H)�H��H��?H��H�H��tH�/H��fD��=/u+UH�=�.H��t H�=�. ��d��.]��w��UH��H��H��]��H�H��Hello, world!0 0: f3 0f 1e fa endbr64 4: 55 push %rbp 5: 48 89 e5 mov %rsp,%rbp 8: 48 8d 05 00 00 00 00lea 0x0(%rip),%rax # f f: 48 89 c7 mov %rax,%rdi 12: e8 00 00 00 00 call 17 17: b8 00 00 00 00 mov $0x0,%eax 1c: 5d pop %rbp 1d: c3 ret In a more complex program, multiple file objects can be linked into one filea.c/a.h -> compile -> a.o b.c -> compile -> b.o a.o b.o -> link -> a.out

Biot-Savart Law

Sat, 18 Jul 2026 02:13:08 GMT

Given a wire of charge and a position , the Magnetic Field at point can be given as
Consider a Current carrying wire and a position P
Since the magnetic field is a Cross Product of the change in the wire length and the vector to , the magnetic field is pointing into the pageConsider a straight finite wire and a point that is on the bisector of that wireBiot-Savart Law Example_0.excalidrawNow consider a situation where is going to infinitySo the magnetic field of an infinite current carrying wire is

AC Steady State

Wed, 03 Jun 2026 22:24:31 GMT

In AC Steady State, the transient responses have died out and the circuit is driven fully by a sinusoidal sourceTo analyze AC steady state circuits Replace all components with its Impedance Analyze the circuit with impedance like a DC non time-dependent circuit
Instead of analyzing resistance in a like in a DC Steady State circuit, we analyze ImpedanceIn AC circuits, filters can cut out waves of certain frequenciesA lowpass filter, as the name suggests, only allows lower frequencies to pass
The voltage of a low pass filter can be calculated by using steady state analysisFind phase of the magnitudeThen thereforeThe frequency response for a low pass filter is shown belowFrequency Response The graph depicts a response in the frequency domain in decibels and not voltage Similar to a lowpass filter, a highpass filter only allows higher frequencies through
Similar to a low pass filter, the voltage of a high pass filter can be calculatedThereforeThe frequency response for a high pass filter in decibels is shown below

Memory Layout

Mon, 01 Jun 2026 00:38:05 GMT

In a C executable, the memory is stored as below

Bayer's Theorem

Fri, 29 May 2026 22:21:35 GMT

Consider a situation where several mutually exclusive events have been determined along with their initial probabilities Now consider an events that changes the probabilities of these events, now the updated probabilities are Bayer's Theorem can be used to update the probabilities as soWhich can be simplified using the Law of Conditional ProbabilityFrom set theoryAnd by substituting the law of conditional probability So the full form of Bayer's Theorem for updating a single probability can be given byFor all and Note that Bayer's Theorem will only work when

Exponential Distribution

Fri, 29 May 2026 21:38:14 GMT

The Exponential Distribution for a Stochastic process is a continuous, non negative distribution specified by one parameter: its Expected Value The probability distribution for a random variable is given byThe expected value is given by The variance is given by The standard deviation is also given by
Unlike both the Normal Distribution and Poisson Distribution, the cumulative distribution can be given as a closed form expressionWhere
Consider a server that processes information with an exponential distribution with an Expected Value of What is the probability that a request gets processed in less than From taking the equation of an exponential distributionAnd substitution our parameters

Electric Field of Continuous Charge

Fri, 29 May 2026 21:23:27 GMT

Where is the distance from to is the vector form to Consider a ring along the axis

Poisson Distribution Example

Fri, 29 May 2026 21:21:34 GMT

Normal Distribution

Fri, 29 May 2026 21:13:07 GMT

A normal or gaussian distribution is a distribution of a real-world random variable specified by two parameters, the Expected Value and the Standard Deviation The normal distribution can be represented via The position of the curve is determined by the The height of the curve is determined by the Probability in a Normal Curve The area of a normal curve is equal the probability of that domain occurring The total area of a normal curve is always equal to

Normal Distribution Example

Fri, 29 May 2026 21:06:06 GMT

Expected Value

Fri, 29 May 2026 20:59:13 GMT

Expected Value is the averaged expected value among all numbered outcomes of Sample SpaceA an expected value of a random variable is often denoted as or The expected value of a common die of six sides can be given as belowHowever, is not rollable number on the die, it just represents the average value of the dice roll as the amount of rolls approach infinity

Event Operations

Fri, 29 May 2026 04:10:18 GMT

Operations can be applied to Events similar to Set OperationsThe union between events and describes the set of outcomes of the events or or bothA union is generally notated as The intersection between events and describes the set of outcomes shared by both and An intersection is generally notated as If the intersection between and is empty, then and are mutually exclusive
The complement of event describes the set of outcomes in the Sample Space that are not the outcomes of A complete is often represented as a hat The union of an event and its complements can be given as so SinceComplements also follow the equality
Where is the entire Sample SpaceConsider three coins being flipped , and let event be at least one heads being flipped Find the followingThe outcomes of is any case where there are at least one heads and at most two tails And the outcomes of is all outcomes that doesn't contain Since contains out of the outcomesThe probability of can be calculated two ways

Event

Thu, 28 May 2026 21:28:21 GMT

An event is a subset of all possible outcomes in a Sample SpaceIt is often denoted as a capital letter after a probability functionThe event contains the sum of the probabilities of all outcomes Consider three coins being tossed with being the probability for heads and being the probability for tailsThe probabilities of these eventsCan be given as such

Basic Probability

Thu, 28 May 2026 21:11:44 GMT

A probability is the change that an outcome will occur from a space of all possible outcomesConsider a coin toss, the probability that heads will land is and that probability that tails will land is also

Faraday's Law Example

Wed, 13 May 2026 18:44:51 GMT

Faraday's Law

Wed, 13 May 2026 17:32:47 GMT

The change in magnetic flux over a closed loop causes an induced voltage as an Electromotive ForceWhere
Negative EMF The EMF generated is negative of the change in magnetic flux because of Lenz's Law Consider two parallel wires with a resistor connecting it and a moving conductive rodConsider a wire with a current and a moving closed loop of wire

Current

Wed, 13 May 2026 17:26:16 GMT

CurrentCurrent is flow of positive charge, it's denoted by ( for fixed currents) Current is measured in Amperes and is a standard SI Unit Flow of positive charge Current is the flow of positive charge while electrons contain negative charge, in reality electrons flow the opposite direction from the current Since charge carriers do not travel in a straight line, drift velocity measures the general velocity of charge carriersThe current density is the measure of the flow of current per unit area and is dependent on drift velocity

Ampere's Law

Wed, 13 May 2026 17:23:48 GMT

The magnetic field along a closed path is proportional to the enclosed currentWhere is the Current Density
Amperean Loop Similar to a Gaussian surface, the loop in which the magnetic field is integrated over is arbitrary Consider a solenoid where the region within the solenoid is a straight magnetic fieldThe inner magnetic field can be found using Ampere's law

Poynting Vector

Wed, 13 May 2026 16:22:22 GMT

The poynting vector conveys the direction and intensity of an Electromagnetic WaveThe direction of the poynting vector is always in the direction of the electromagnetic wave no matter the phase

The time average of the Poynting vector is the intensity of a WaveWhere is the intensity And where is the power And where is the area

Poynting Vector Example

Wed, 13 May 2026 16:20:37 GMT

Capacitance

Wed, 13 May 2026 16:05:15 GMT

Capacitance is a measure of potential charge given in units of faradsThe capacitance of a system only depends on the geometry of the system.To find the potential of conductorTherefore the equations of the potential across a conductorFor capacitors in seriesFor capacitors in parallelConsider two parallel plates of area and charge that are distance away
The Electric Potential of the two plates can be found via Consider bringing in charges from infinity, the work needed to bring the first charge is zero The work needed to bring the second charge is
Generally it can be given asConverting to energyAnd since
Where is the energy stored The energy within a capacitor is stored within the Electric Field
Consider two spherical shells
The time it takes to charge a capacitor is proportional to the total resistance and capacitance in the circuitUsing , we can find the charge of a capacitor as function of time

Electric Potential

Tue, 12 May 2026 22:28:55 GMT

The electric potential is the energy held by a charge due to surrounding Electric FieldsFor a point chargeElectric potential has units of or or voltsIn mechanics, work is dependent on displacementWorkWork is defined as the amount amount of force applied over a distanceTherefore the total work done over a distance is equal to the change in kinetic energy between the two distances Potential Energy Kinetic EnergyIn a system where energy is conserved
For Conservative Forces, the path of the object does not affect the total work done by the force Since electric fields are conservative, the work is equal to the total displacement done by electric fieldsThe change in potential energy of given work isThen the change electric potential given the potential isPotential vs Electric Potential Potential energy and electric potential energy is not the same, while potential energy behaves similarly to mechanics, electric potential energy depends on the charge of the object Find the electric potential between two charged spherical shells
The electric potential and the electric field has the relation ofMeaning the electric potential can be found from the electric field and vise versa
Find the electric potential of a source then use it to solve for the Electric Field
Find the electric potential of a uniformly charged rod

Magnetic Dipole

Tue, 12 May 2026 21:52:06 GMT

Magnets usually have a north and a south pole which is called the dipoleThe magnetic field over an enclosed area must be zero Therefore, magnetic monopoles can not existThe the magnetic dipole moment is the direction and strength of a the magnetic field from a source that induces a magnetic field The magnetic dipole moment is defined as

Coulomb's Law

Tue, 12 May 2026 21:12:53 GMT

Coulomb's Law represents the Electric Field created by two charges

Node Analysis Example_0

Mon, 11 May 2026 05:35:34 GMT

Serotonin

Mon, 11 May 2026 04:20:19 GMT

NH₂NHHO

Oxidation Numbers

Mon, 11 May 2026 04:16:52 GMT

Formal Charge and oxidation numbers are two extremes of election assignmentWhen a bond splits homolytically, the electrons are hypothetically split apart evenly so each atom receives one electron. This gives a formal charge of for both atoms.When a bond splits heterolytically, one atom, usually the more electronegative one receives both electrons. This gives a oxidation charge of and The actual charge is intermediate between the two extremes, both of these models are hypothetical representations.homolytically: splits evenly heterolytically: splits unevenly, favoring one side Fluorine always has an oxidation number of Oxygen usually has an oxidation number of Hydrogen has an oxidation number of when bounded to nonmetals but when bounded to metals The sum of the oxidation numbers is the charge on a molecule.Find the charge of is assigned The molecule has a charge of

DC Steady State

Sat, 09 May 2026 20:36:28 GMT

In time dependent circuit, a steady state is achieved after a significant amount of time when there is no longer a change in voltage or currentTo analyze DC steady state circuits Replace all Inductors with open circuits
Replace all Capacitors with short circuits Analyze the resulting circuit like a non time-dependent circuit In a steady state circuit the Consider this time dependent circuit
After converting all time-dependent components into their steady state equivalents, we get this circuitNow solving for the remaining systemAnd since the remaining system is in steady state (no voltage or current changes)The capacitor has no current (open circuit) and the voltage of And the inductor has no current (short circuit) and the current of Because this system has both a capacitor and an inductor we need to take into account two time constantsSince the circuit reaches steady state in approximately , we take the longer value

DC Steady State Example_0

Sat, 09 May 2026 18:08:40 GMT

DC Steady State Example

Sat, 09 May 2026 18:08:31 GMT

Capacitor

Fri, 08 May 2026 04:33:49 GMT

Capacitors store charge between its two plates and stores energy between the plates via an Electric FieldThe capacitor is given by this symbolThe capacitance is read from a three-digit code XYZ where: X and Y are significant figures Z is the multiplier (power of 10 to multiply by) Examples 103K = 10 × 10³ pF = 10,000 pF = 10 nF, ±10% tolerance 105 = 10 × 10⁵ pF = 1,000,000 pF = 1 µF (tolerance typically ±20% if not specified) 221 = 22 × 10¹ pF = 220 pF The charge stored in a capacitor can be determined via
Where is the Capacitance of the capacitorAnd as Capacitors store energy equal to one half of the conductivity of the inductor times voltage squared
This energy is usually stored within a dielectric as a Electric Field
Unlike, Resistors, capacitors in series divide all voltage across all capacitorsVoltage and Current While resistors in parallel share a voltage split current, capacitors in series will usually share a common current and split voltage Capacitors in parallel combine their capacitance
When paired with a resistor in a RC Circuit, a capacitor charges and discharges according to the time constantA capacitor reaches approximately 63% of its final voltage after one and is considered fully charged after

Capacitor Example

Fri, 08 May 2026 04:23:43 GMT

Operational Amplifier

Fri, 08 May 2026 04:11:37 GMT

An operation amplifier (Op Amp) takes in a positive and negative voltage then outputs a multiple the difference of the two inputsThe op amp is drawn as a triangle, the voltage source is often emitted The op amp does not draw current, instead, it uses a outside voltage sourceThe gain is often large but never exceeds the external power supplyThe gain of the Op Amp is heavily variable, it subject to changes in temperature, humidity, and background conditionsConsider this op-amp circuitIf is slightly higher than then will be Vice versa, if is slightly lower than then will be Consider the system
Component multiples input into while component takes a fraction of and feeds it into the negative of the input sumThe system can be modeled asSo to double the input , must be In electronics, an op amp can be used to double voltage as follows
For Op Amps with feedback, if the voltage then there is no gain or
This is called a virtual shortZero Voltage If the voltage is zero, then the voltage output is not zero, it would be solved differently than Consider a sensor and a speaker output with a
Because the sensor will often not have enough voltage to power the speaker An op amp can be used to amplify current in a circuit known as the Unity Gain Circuit
The voltage can be model as followsBecause And therefore the input of the sensor is unchanged while the current is amplifiedIf a system is linear, an input signal will produce equivalent output signal However, for non-linear systems, the output will not be equivalent
The op amp can filter distortion by using a feedback loop
An op amp can be tested for negative feedback using the following process Remove op amp from circuit Set all voltage and current sources to zero Place a voltage source at op amp's former output node If any part of the that voltage gets to the input's negative terminal, then there is negative feedback Consider this circuit
Since the voltage of the test source reaches the negative terminal The circuit has a feedback loopNow consider this circuit
While the current reaches the negative terminal, the negative terminal is connected to ground so therefore no voltage reaches the negative terminal The circuit does not have a negative feedback loopNow consider this cat
Isn't he silly.Finally, consider this circuit
Since neither the current nor the voltage reaches the negative input terminal The circuit does not have a negative feedback loopThere are two types of amplifiers using Op Amps, inverting and noninvertingNoninverting amplifiers amplify the signal voltage in phase with the input
Where the amplification factor is a function of the resistorsVoltage Follower A buffer can be created when is close to 0 and is close to infinity (or removed) Inverting amplifiers amplify the signal voltage opposite of the phase of the input
Where the amplification factor is a function of the resistors

Operational Amplifier Example_13

Fri, 08 May 2026 04:08:12 GMT

Operational Amplifier Example_12

Fri, 08 May 2026 03:59:07 GMT

Operational Amplifier Example_11

Fri, 08 May 2026 03:40:01 GMT

Thevenin and Norton Theorems Example_1

Thu, 07 May 2026 21:10:20 GMT

Thevenin and Norton Theorems Example

Thu, 07 May 2026 20:58:32 GMT

Thevenin and Norton Theorems Example_0

Thu, 07 May 2026 20:58:19 GMT

Node Analysis

Thu, 07 May 2026 19:49:58 GMT

Node Analysis

Thu, 07 May 2026 19:49:55 GMT

Node Analysis is a set of procedures to solve simple circuitIt's steps are as follows: Identify all Nodes Chose one node as a reference Asking variable names to the rest of the nodes
Write KCL at each unknown node Solve resulting linear system for the the voltages Use the voltages to solve for any needs currents Consider a system of resistors
Using these equalities, solve for voltages via linear algebraThen use voltages along with KCL to solve for currents

Sockets

Sun, 03 May 2026 06:51:06 GMT

Sockets are channels of communication between two endpointsSome types of sockets includes Internet Sockets, UNIX Sockets, and sockets over Bluetooth, Radio, or other mediums of informationUDP is a single socket method of communication that's analogous to postal mail. While the size of information transmitted might be high, there is no guarantee of delivery, and the route of each packet might be differentTCP is a single socket method of communication that's analogous to a telephone call. The size of each individual packet is small but the stream is continuous Sockets can be identified by a combination of their host IP Address and Port NumberWhile multiple services can exist within a host, only one process can occupy a port number at a timeThe connection of a socket follows a series of protocols starting with the socket function The socket function initializes the socket connection between the hostsint socket(int domain, int type, int protocol)Where domain is the protocol family AF_INET for IPv4 AF_INET6 for IPv6Where type is the semantic of the communication SOCK_STREAM for TCP SOCK_DGRAM for UDPWhere protocol is the specific protocol and is usually 0After the socket() is established the connection can be bound via bindint bind(int sockfd, const struct sockaddr *addr, socklen_t addrlen)Where sockfd is the socket identifier, the return value of socket() Where sockaddr is the protocol specific struct Where addrlen is the size of the address in bytesByte Ordering With little endian, the least significant byte is stored at the lowest address. With big endian, the least significant byte is stored at the highest address. While LINUX uses little endian, TCP/IP uses big endian After the bind(), the listen function can be called to mark the server as a passive server, a passive server, or a server that can accept incoming connections via acceptint listen(int sockfd, int backlog)The accept function waits for a request to come in, this blocks the process until a request arrivesint accept(int sockfd, struct sockaddr *addr, socketlen_t *addrlen)

Process

Sun, 03 May 2026 03:50:49 GMT

A process is a executable unit that handles a single programA zombie process is a process where termination of the process has not been handledA zombie process can be created by failing to wait before the child process is finishedAn orphan process is a process where the parent process no longer existsThis can happen is the parent process exits before the child process is finished or if the parent process does not wait for the child processOrphans are created when their parents die (either through killing them directly or by natural causes). Because their parents do not exist, orphans are useless entities that can take up space and resources within the system at large while contributing nothing.It is then important to kill orphans as soon as possible to save on resources. Orphans can be killed in any number of ways.The most popular way to kill orphan is via killkill -9 [Name of Orphan] Below is a orphan killing algorithm#!/bin/bash # Kill orphan processes: user-space processes re-parented to init (PPID=1) GRACE=5 MIN_AGE=60 while IFS= read -r pid comm; do [[ ! -s "/proc/$pid/cmdline" ]] && continue uid=$(awk '/^Uid:/{print $2}' "/proc/$pid/status" 2>/dev/null) [[ "${uid:-0}" -le 999 ]] && continue age=$(( $(date +%s) - $(stat -c %Y "/proc/$pid" 2>/dev/null || echo "$(date +%s)") )) [[ "$age" -lt "$MIN_AGE" ]] && continue echo "Killing orphan: PID=$pid CMD=$comm UID=$uid AGE=${age}s" kill -TERM "$pid" 2>/dev/null sleep "$GRACE" kill -0 "$pid" 2>/dev/null && kill -KILL "$pid" 2>/dev/null done < <(ps -eo pid=,ppid=,comm= | awk '$2 == 1 && $1 != 1 {print $1, $3}')

Protocol

Sun, 03 May 2026 01:37:29 GMT

A protocol is an agreement between two parties on a method of communication

Signal

Sun, 03 May 2026 01:36:33 GMT

A signal is a type of interrupt that change the flow of a program, they can communicate hardware level problems (such as division by 0), OS level exceptions (such as alarm) and process level requests (such as kill)Signals can be generated via functions such as the alarm and kill functionsunsigned int alarm(unsigned int seconds) sets and alarm and sends the SIGALRM signalint kill(pid_t pid, int sig) sends the specified signal to the process with the pidSignals can be handled with both signal and signactionWhen a signal is called, the operating system stops the program then passes the signal to the signal handler, the handler then processes the signal and returns control back to the programSignals used to be handled via the signal handler#include #include #include void sighandler(int signo) { if (signo == SIGINT) { printf("Received SIGINT\n"); } } int main() { if (signal(SIGINT, sighandler) == SIG_ERR) { printf("Can't catch SIGINT\n"); } // A long sleep so we can issue a signal. while (1) { sleep(1); } return 0; } However, this way of handling lacks the ability to block other signals from being called during the handle process, signal also have different behavior on different systemsModern programs use sigaction which changes the way the process operates The structure of the function call is as belowsint sigaction(int signum, const struct sigaction *act, struct sigaction *oldact) Where signum can be any signal except SIGKILL or SIGSTOPstruct sigaction { void (*sa_handler)(int); void (*sa_sigaction)(int, siginfo_t *, void *); sigset_t sa_mask; int sa_flags; void (*sa_restorer)(void); } The mask sa_mask is the set of signals to block while the signal is being handled, the signal that triggered the handler is blocked by default#include #include #include void handle(int sig, siginfo_t *siginfo, void *context) { printf("Sending PID: %ld, UID: %ld\n", (long)siginfo->si_pid, (long)siginfo->si_uid); } int main() { struct sigaction act = {0}; // Use the sa_sigaction field because the handle has two additional // parameters. act.sa_sigaction = handle; // The SA_SIGINFO flag tells sigaction() to use the sa_sigaction field, not // sa_handler. act.sa_flags = SA_SIGINFO; if (sigaction(SIGTERM, &act, NULL) < 0) { perror("sigaction"); return 1; } while (1) { sleep(1); } return 0; } A reentrant function is function that can safely be interrupted and restarted without causing issues To be a reentrant function, the function must satisfy the following It may not use global/shared resource It should operate only with the data provided by the caller It must call only other reentrant functions

Lorentz Transformation Example_1

Tue, 28 Apr 2026 14:41:33 GMT

Lorentz Transformation

Tue, 28 Apr 2026 14:34:35 GMT

Lorentz Transformation Example_0

Tue, 28 Apr 2026 14:33:54 GMT

Lorentz Transformation Example

Tue, 28 Apr 2026 14:26:56 GMT

Electromagnetic Wave

Tue, 21 Apr 2026 15:08:23 GMT

As a wave travels, the Electric Field and the Magnetic Field are orthogonal to the wave and each other
The speed of the wave is equal to The wavelength of the wave is equal to Consider Faraday's Law

Electromagnetic Wave Example

Tue, 21 Apr 2026 14:40:04 GMT

Self Inductance

Tue, 14 Apr 2026 14:47:11 GMT

Coils with a current source induces Magnetic Fields that then induces a current when that current source is off

Self Inductance Example

Tue, 14 Apr 2026 14:37:50 GMT

Mutual Inductance

Tue, 14 Apr 2026 14:32:59 GMT

A Magnetic Field induced from a Current in one coil induces a magnetic flux within another, coupled coilConsider two concentric circles with current flowing through the other coil

Mutual Inductance Example_0

Tue, 14 Apr 2026 14:29:21 GMT

Mutual Inductance Example

Tue, 14 Apr 2026 14:24:53 GMT

Linear Independence

Tue, 07 Apr 2026 18:20:25 GMT

A set vectors can be can be determined independent by using its corresponding Linear FunctionFor a set of vectorsIf only has a trivial solution then all vectors are linearly independentIf has non trivial solutions, then the vectors are linearly dependentAlternatively, the vectors in form must have a trivial solutionConsider the vectorsTo determine linear independency, must be a trivial solution
To show this, we can turn into a upper triangular matrix in Echelon formBecause has pivots for a matrix then is a trivial solutionTherefore, vectors ,, and are all linearly independent

Matrix Factorization

Tue, 07 Apr 2026 18:05:11 GMT

A matrix factorization expresses a Matrix as a product of two or more matricesLU Factorization is a kind of matrix factorization that uses two matrices and Where is A lower triangular matrix With 's on its diagonal Where is A upper triangular matrix
LU Theorem Assume is an matrix that can be reduced to Echelon form without Row Interchange Then, can be uniquely written in the form Consider a matrix
Using Elementary Matrix Operations, reduce into Echelon form (without interchange)
Pivot Row Like in Gaussian Elimination, the pivot row should always be used first when performing replacement operations Failing to do so may not generate a upper triangular matrix This upper triangular matrix is now
can then be represented as a a product of Elementary Matrices
By taking the Inverse all of the matrices, we can rewrite Then the inverses becomes Then finally, we can write as soWhen factorizing a matrix into an matrix, the interchange and scaling operations are not allowedIf interchange is necessary, then a helper matrix can be used to transform into a matrix that does not need scalingThen A can be represented as soScaling When scaling, the matrix, by definition, can not have the same identity No matrix should necessitate scaling during factorization

Geometrical Model for Gauss Jordan

Tue, 07 Apr 2026 18:05:11 GMT

The Gauss-Jordan Elimination can be solved geometrically via using vectorsConsider a two equation systemAfter graphing the system, we can determine there is only one solution for the systemA system is made out of equations and unknown variables Theses systems can be consistent or inconsistent based on their solutionsConsider the system and it's matrix equivalentConsider the pivot column It's variable is called the basic variable while the other variables , are called free variables
pivot: Leading entry in Echelon form The solution to the system can have multiple formsHowever, generally, we would have to solve for the basic variable in terms of free variablesConsider the systemWhen constructing equations, ensure basic variables are solvedSo that for the equations in matrix form, the basic variables are Consider the systemBecause the solution is a pivot, there are no solutions There is not enough equations
In a reduced Echelon form of If a row matching Then the system has no solutions and is inconsistent

Inverse Elementary Matrix Operations

Tue, 07 Apr 2026 17:32:53 GMT

All Elementary Matrix Operations can be inversed
For a Identity Matrix An inverse replacement operation subtracts values from one row to anotherAn inverse interchange operation behaves exactly the sameAn inverse scaling operation scales the middle value of the matrix inversely

Elementary Matrix Operations

Tue, 07 Apr 2026 17:32:48 GMT

Elementary matrix operations are simple operations that can be applied to matrices All elementary operations are Inversible
For a Identity Matrix An replacement operation adds values from one row to anotherAn interchange operation swaps one row for anotherA scaling operation scales the middle value of the matrix

Memory Errors

Mon, 06 Apr 2026 15:35:39 GMT

Memory errors are errors relating to management of Memory within a C ProcessOccurs when a memory is read from a location that does not existSimilar to invalid read but occurs when memory is written to a location that does not existThe message "conditional jump or move depends on uninitialized value(s)" occurs when the operation depends on a variable that is uninitializedconditional jump: a statement that controls flow such as if, for, or while statements move: any other kind of read from memoryA segmentation happens when the operation system intervenes before the program crashes, among other things, a common cause of it is dereferencing a null pointer

Faraday's Law Example_0

Fri, 03 Apr 2026 00:22:03 GMT

Echelon

Thu, 02 Apr 2026 18:51:54 GMT

An echelon or an echelon form is a Matrix All rows with zeros are below nonzero rows All nonzero leading entries must be below and to the right of all others All entries below a leading entry must be zeros Below are examples of echelon form matricesThe identity matrix is in echelon formEverything right and above the leading entries can be any number

Pasted image 20260402142806

Thu, 02 Apr 2026 18:28:06 GMT

Gaussian Elimination

Thu, 02 Apr 2026 17:52:48 GMT

Gaussian Elimination is a process in which a system of equations can be solved via solving the system as a Linear SystemConsider a system of equationsWe can take turn all coefficients and into an augmented matrix consisting of a coefficient matrixThen use multipliers of the first row to eliminate the the first coefficient from the first column The coefficient matrix can be modified to reduce unknowns while keeping the matrix identity the sameThis operation can be repeated againFrom here, the term is isolated and can be solvedThe whole equation can be solved via backwards substitution or via additional eliminationsIn gaussian elimination, the order of elimination goes as followsThe Gauss Jordan Model is a modification of the Gaussian Elimination where the resulting coefficient matrix is an identity matrixInstead of a Gaussian Elimination simply eliminating a coefficientThe Gauss-Jordan model reduces that coefficient into a Consider the system
By the Distributive PropertyThe same product can be found via a single row operation of the Gaussian Elimination
This means that for any matrix , its inverse form can not exist if it can not follow the Identity Property

Gaussian Elimination Example_2

Thu, 02 Apr 2026 17:40:47 GMT

Gaussian Elimination Example_1

Thu, 02 Apr 2026 17:31:01 GMT

Gaussian Elimination Example_0

Thu, 02 Apr 2026 17:29:40 GMT

Gaussian Elimination Example

Thu, 02 Apr 2026 17:16:34 GMT

Lenz's Law

Thu, 02 Apr 2026 14:44:55 GMT

Lenz's Law is made out of two parts that deal with forces generated by changes Induced EMF will always be in the direction that opposes the direction of the change in magnetic flux that caused it Induced currents, torque, and force will always oppose the direction of the change in magnetic flux that caused it

Lenz's Law Example

Thu, 02 Apr 2026 14:34:48 GMT

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Linear System

Tue, 31 Mar 2026 18:03:08 GMT

A linear system of linear algebraic equations is a set of linear equations without differentialsWhere the system can be given as a MatrixComposed of unknown vector and constant vectorsSo that the system can be given asWhere is the coefficient matrix and are unknown vectors

Electromotive Force Example

Tue, 31 Mar 2026 14:57:10 GMT

Electromotive Force

Tue, 31 Mar 2026 14:42:24 GMT

The electromotive force (EMF) is the generalized force per unit charge along a closed pathThe force is measured in voltsConsider a closed loop of wire partially in a Magnetic Field

We can use Faraday's Law to calculate the change in magnetic flux, which is equal to the EMF

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Gate X

Mon, 30 Mar 2026 20:22:50 GMT

The X gate, similar to a NOT Gate, denoted as X is a gate that flips the spin of the qubit

Switch

Mon, 30 Mar 2026 20:22:01 GMT

A switch is a type of component that can start or stop the flow of currentA switch has different symbols depending on its state

Switch Example

Mon, 30 Mar 2026 20:21:39 GMT

String

Sat, 28 Mar 2026 19:22:35 GMT

Strings in C are represented by arrays of characters that ends with a \0 or a null terminatorA string requires bytes where is the amount of charactersCommon string functions includeint strlen(const char *s) which returns the length of the string char *strcpy(char *t, const char *s) which copies string into an already allocated array char *strdup(const char *s) which returns a heap copy of the string

Heap

Sat, 28 Mar 2026 19:21:52 GMT

The heap is a section of memory managed directly by the programAs the name suggests the memory is stored as a HeapThe malloc() function reserves a section of the heap by the amount of bytes provided and returns a pointer to the beginning of the block The free() function frees a section of memory by taking the pointer of the first block as input

Ampere's Law Example_0

Sat, 28 Mar 2026 19:19:48 GMT

Command Line Argument

Sat, 28 Mar 2026 19:17:18 GMT

The C main program has two command line argument variablesint argc number of command line arguments char **argv command line arguments in a String Array

Keywords

Sat, 28 Mar 2026 19:11:39 GMT

Keywords are words that have functional properties and can not be used as variablesThere are as follows

Maxwell's Equations

Thu, 26 Mar 2026 15:19:27 GMT

Maxwell's Equations are a set of equations that describes the relationship between the Electric Field and the Magnetic Field
Gauss's Law
Magnetic Flux Law
Ampere's Law
Faraday's Law Gauss's Law Magnetic Flux Law Ampere's Law Faraday's Law

Ampere's Law Example

Thu, 26 Mar 2026 14:53:55 GMT

Fork Pathway

Wed, 25 Mar 2026 22:26:40 GMT

A fork pathway is a way to visualize forks within a programConsider the functionvoid function1() { pid_t pid; if ((pid = fork()) == 0) { sleep(1); printf("C1 "); fflush(stdout); } if ((pid = fork()) == 0) { sleep(2); printf("C2 "); fflush(stdout); } if ((pid = fork()) == 0) { sleep(3); printf("C3 "); fflush(stdout); } // Small hack to wait for all descendent processes while (wait(NULL) != -1) { } } The Fork Pathway for this function would look like the following:

Kernel Call Exec

Wed, 25 Mar 2026 22:20:49 GMT

The exec kernel call changes the current process into a different programThe execl family takes in a list of elements, the first argument being being the file path and subsequent arguments being the argument list execl("cat", "cat", "/src/a.txt"); Program name repetition The program name is repeated twice in execl due to different program paths and alias'

Fork Pathway Example

Wed, 25 Mar 2026 22:15:40 GMT

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Matrix Multiplication

Tue, 24 Mar 2026 18:51:27 GMT

Two matrices can be multiplied together

Matrix Transposition

Tue, 24 Mar 2026 18:40:30 GMT

Matrices can be transposed, where the Row Vector become the Column Vector and vice versaA transposition is reversable, meaning two transposition operations result in the same matrix

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Matrix Exponents

Tue, 24 Mar 2026 18:39:24 GMT

Matrices can be have exponentsWhereMatrices can have an inverse form whereWhere is the Identity MatrixLet , be integers

Scaling Matrix

Tue, 24 Mar 2026 18:16:28 GMT

A fixed matrix can be "scaled" via using a scaling matrixConsider these examples of scaling matricesGet the same matrix Pick up the first row of B and put on the first row of C Pick up the first row of B put on the second row of C Pick up the second row of B and put on all rows of C Put the sum of each column on the second row of C Keep the first row of B, multiply the second row of B by 2, and divide the third row of B by 2

Inductor Example

Mon, 23 Mar 2026 17:46:29 GMT

Gate Z

Wed, 18 Mar 2026 21:15:34 GMT

The Z Gate, denoted as Z, flips the sign of the value and leaves the value unchanged

Gate Hadamard

Wed, 18 Mar 2026 16:59:15 GMT

The Hadamard gate, denoted as H, creates a superposition where all probabilities are equally likely

Qubit

Wed, 18 Mar 2026 16:44:40 GMT

A qubit is the information stored in the spin of an atomIn classical computation, a bit can be either 0, or a 1 However, in quantum computation, the bits are measured as down spin or up spin Dirac Notation Instead of a discrete bits, quantum uses Dirac notation to measure the information of the bit

Gauss's Law

Thu, 12 Mar 2026 15:19:27 GMT

The flux through a surface is equal to the enclosed charge over Gauss's Law can be applied to uneven surfaces since it only relies of volume enclosedConsider a spherically symmetrical charge configuration

Current Loop in External Magnetic Field

Thu, 12 Mar 2026 15:03:06 GMT

Consider a loop of Current in an external Magnetic Field
The force of the magnetic field on the wire can be given byThe force can be broken up into four legs whereSo that

Current Loop in External Magnetic Field Example

Thu, 12 Mar 2026 14:54:37 GMT

Magnetic Dipole Example

Thu, 12 Mar 2026 14:38:08 GMT

Biot-Savart Law Example_1

Wed, 11 Mar 2026 18:10:40 GMT

Biot-Savart Law Example_0

Tue, 10 Mar 2026 14:39:58 GMT

Biot-Savart Law Example

Tue, 10 Mar 2026 14:24:41 GMT

Magnetic Field Example_0

Thu, 05 Mar 2026 16:12:40 GMT

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Magnetic Field Example

Thu, 05 Mar 2026 15:46:12 GMT

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Higher Order Linear

Thu, 26 Feb 2026 19:21:44 GMT

Consider an equation of higher order derivativesIf and are continuous on an open internal , , then there exists a unique solution to both equations and the solution is defined throughout .

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Classification of Differential Equations

Thu, 26 Feb 2026 18:12:34 GMT

Differential Equations can be classified in many different waysSolutions to Ordinary Differential Equation only have one dependent variable while Partial Differential Equations have two or moreDifferential equations can be given in a single equation or a set of equationsThe order of different equations refer to the highest derivative involvedFor example,Would be a 2nd order partial derivative Differential equations can be classified as linear if all of derivatives of the unknown term are linearAny non-linear component makes the equation non-linear

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Resistor

Thu, 26 Feb 2026 16:09:18 GMT

Resistors resists charge within a circuit by making it harder for positive charge to flow throughResistors in series combine their resistanceResistors in parallel divide the current between all resistorsJustification for resistors in parallelIf then If then

Electrical Meter

Thu, 26 Feb 2026 16:07:19 GMT

Meters are instruments that measures a value in real timeAn ammeter measures current in units of mA, it has an in side and an out side and measures the current from the in side to the out sideA voltmeter measures voltage in units of V, it measures voltage form its positive side to the negative side Reverse Current If a voltmeter is supplied with current that travels from the negative side to the positive side, the voltage is reversed An Ohmeter measures resistance by measuring the voltage of a standard voltage source

Circuit Analysis

Thu, 26 Feb 2026 16:07:19 GMT

The current and voltage of components and Nodes within a circuit can be identified using Kirchhoff's Laws and Ohm's LawBecause of KCL, the sum current in into a system of resistors is the same a the sum current out of the system Current of a resistor in parallel $$ i_{a}=i_{\text{total}} \frac{R_{\text{parasitic}}}{R_{a}} $$

DC and RC Circuits Example_0

Thu, 26 Feb 2026 16:00:09 GMT

DC and RC Circuits Example

Thu, 26 Feb 2026 15:37:26 GMT

Ohm's Law Example

Thu, 26 Feb 2026 15:33:36 GMT

Current

Thu, 19 Feb 2026 16:20:50 GMT

Current is flow of positive charge, it's denoted by ( for fixed currents) Current is measured in Amperes and is a standard SI Unit Flow of positive charge Current is the flow of positive charge while electrons contain negative charge, in reality electrons flow the opposite direction from the current

Current Example_0

Thu, 19 Feb 2026 16:15:43 GMT

Current Example

Thu, 19 Feb 2026 16:08:33 GMT

Capacitance Example_2

Thu, 19 Feb 2026 16:01:22 GMT

Capacitance Example_1

Thu, 19 Feb 2026 15:43:57 GMT

Capacitance Example_0

Thu, 19 Feb 2026 15:34:08 GMT

Capacitance Example

Thu, 19 Feb 2026 15:26:22 GMT

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Amplification with Op Amps Example_3

Wed, 18 Feb 2026 22:36:49 GMT

Amplification with Op Amps Example_5

Wed, 18 Feb 2026 22:27:08 GMT

Amplification with Op Amps Example_4

Wed, 18 Feb 2026 22:24:38 GMT

Amplification with Op Amps Example_1

Wed, 18 Feb 2026 22:10:04 GMT

Amplification with Op Amps Example_2

Wed, 18 Feb 2026 22:08:14 GMT

Amplification with Op Amps Example_0

Wed, 18 Feb 2026 21:58:00 GMT

Amplification with Op Amps Example

Wed, 18 Feb 2026 21:43:50 GMT

Operational Amplifier Example_9

Wed, 18 Feb 2026 21:27:09 GMT

Operational Amplifier Example_10

Wed, 18 Feb 2026 21:27:04 GMT

Operational Amplifier Example_8

Wed, 18 Feb 2026 21:25:01 GMT

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Homogenous Equations with Constants Example_5

Tue, 17 Feb 2026 18:59:12 GMT

Homogenous Equations with Constants Example_0

Tue, 17 Feb 2026 18:49:19 GMT

Homogenous Equations with Constants Example_4

Tue, 17 Feb 2026 18:20:32 GMT

Conductor and Insulators Example_1

Tue, 17 Feb 2026 15:44:08 GMT

Priority Queue

Mon, 16 Feb 2026 23:01:29 GMT

A priority queue is a type of Queue where instead of the last element, the element with the most priority is removed first, it supports the following methodsinsert(x):
(Linked Lists)
(Red-Black Tree) deleteMin():
(Linked Lists)
(Red-Black Tree)
However, a better time cost can be achieved by using a COMS Data Structures/Heap getMin(x): (Deletion is ) insert(x): sort():
(Linked Lists)

Operational Amplifier Example_2

Mon, 16 Feb 2026 22:20:51 GMT

Operational Amplifier Example_7

Mon, 16 Feb 2026 22:20:34 GMT

Operational Amplifier Example_6

Mon, 16 Feb 2026 22:11:44 GMT

Operational Amplifier Example_5

Mon, 16 Feb 2026 22:08:05 GMT

Operational Amplifier Example_4

Mon, 16 Feb 2026 21:58:50 GMT

Operational Amplifier Example_3

Mon, 16 Feb 2026 21:56:28 GMT

Operational Amplifier Example

Mon, 16 Feb 2026 21:48:08 GMT

Operational Amplifier Example_0

Mon, 16 Feb 2026 21:44:15 GMT

Operational Amplifier Example_1

Mon, 16 Feb 2026 21:43:44 GMT

Node Analysis Example

Sun, 15 Feb 2026 20:18:00 GMT

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Electric Field Example_0

Fri, 13 Feb 2026 01:25:44 GMT

Electric Potential Example_0

Fri, 13 Feb 2026 00:56:31 GMT

Homogenous Equations with Constants Example_3

Thu, 12 Feb 2026 19:13:04 GMT

Homogenous Equations with Constants Example_2

Thu, 12 Feb 2026 18:35:00 GMT

Conductor and Insulators Example_0

Thu, 12 Feb 2026 16:24:15 GMT

Conductor and Insulators Example

Thu, 12 Feb 2026 16:00:31 GMT

Equipotential Contours

Thu, 12 Feb 2026 15:53:49 GMT

Equipotential Contours is a map of contour lines that acts similarly to a height map They follow the following rules: Each point on a specific contour line has the same Electric Potential
The closer the controller lines are to each other the bigger the charge in Electric Potential for a radius
The direction of the Electric Field points form high to low potential is perpendicular to the contour lines

Equipotential Contours Example_1

Thu, 12 Feb 2026 15:51:04 GMT

Equipotential Contours Example_0

Thu, 12 Feb 2026 15:49:52 GMT

Equipotential Contours Example

Thu, 12 Feb 2026 15:49:35 GMT

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Curl of an Electric Field

Thu, 12 Feb 2026 15:29:32 GMT

If the curl of an Electric Field is
Then is ConservativeThis means that the electric field can not loop onto itself

Homogenous Equations with Constants Example_1

Tue, 10 Feb 2026 19:10:00 GMT

Homogenous Equations with Constants Example

Tue, 10 Feb 2026 18:49:15 GMT

Second Order Linear Example_1

Tue, 10 Feb 2026 18:28:10 GMT

Geometry of Electric Fields

Tue, 10 Feb 2026 16:18:29 GMT

Consider Gauss's Law
By using Divergence Theorem, we can rewrite the law into another formThis is considered the Differential Form of Gauss's LawConsider the equation
Recall that is a Conservative Force

Electric Potential Example_1

Tue, 10 Feb 2026 15:40:20 GMT

Type Casting

Mon, 09 Feb 2026 23:04:05 GMT

A cast is implicit when the complier auto casts a value while an explicit cast is called manually within the codeA cast is considered a promotion when the new type more specified than the old, an int is a very primitive type while a float is more specific

Coding Style

Mon, 09 Feb 2026 22:51:29 GMT

Different stylings for C can be used based on different best practices and standardsFor system's programming, most people follow the Linux Kernel Style

Alternating Current

Mon, 09 Feb 2026 21:32:26 GMT

Circuit Analysis Example_0

Mon, 09 Feb 2026 21:24:55 GMT

Circuit Analysis Example

Mon, 09 Feb 2026 21:21:53 GMT

Second Order Linear Example_0

Thu, 05 Feb 2026 19:25:21 GMT

Second Order Linear Example

Thu, 05 Feb 2026 19:00:26 GMT

Separable Equation

Thu, 05 Feb 2026 18:45:15 GMT

An Ordinary Differential Equation can be called separable if it can be written in this form Separable equations can be separated into two parts and can be integrated directly
Exactness All separable equations are Exact This is a separable equation since can be multiplied on both sidesThis is a separable since can be divided (given certain conditions) on both sides This is not separable however since can not be properly removed to fir the separable form

Integral Curve

Thu, 05 Feb 2026 18:45:15 GMT

Curves that map the solutions to an Ordinary Differential Equation

First Order Linear

Thu, 05 Feb 2026 18:45:15 GMT

First Order Linear Ordinary Differential Equation can be solved using either the Method of Integrating Factor or Method of VariationSome equations can have special forms and can be solved generally
These equations are ExactType one equations usually follow the format and can be easily integratedType two equations follow the format and can be reduced to a type oneSingle Separatable Type II

Exact Equation

Thu, 05 Feb 2026 18:36:24 GMT

Given an equation, if there exists functions such that Then the equation is an exact equationThe total derivative of a function can be givenConsider the equationAssume can be given as
By taking the Partial Derivatives, we can see that is equal to the original equationBy equating the partial to we can see that is constantRather than finding and to determine exactness, the following process can be usedLetThen take the complementary derivative of both equationAssume is continuous and differentiable If this condition is satisfied then the equation is exactTo determine exactness

Exact Equation Example_1

Thu, 05 Feb 2026 18:36:15 GMT

Electric Field Example

Thu, 05 Feb 2026 16:03:48 GMT

Electric Potential Example

Thu, 05 Feb 2026 15:55:30 GMT

Work

Thu, 05 Feb 2026 15:17:15 GMT

Work is defined as the amount amount of force applied over a distanceTherefore the total work done over a distance is equal to the change in kinetic energy between the two distances Potential Energy Kinetic EnergyIn a system where energy is conserved

Resistor Example_6

Wed, 04 Feb 2026 22:17:13 GMT

Resistor Example_4

Wed, 04 Feb 2026 21:48:07 GMT

Resistor Example_5

Wed, 04 Feb 2026 21:38:50 GMT

Gauss's Law Example

Wed, 04 Feb 2026 20:09:57 GMT

Kirchhoff's Laws Example

Wed, 04 Feb 2026 19:51:50 GMT

Kirchhoff's Laws Example_0

Wed, 04 Feb 2026 19:47:29 GMT

Basics of Bash Scripting

Mon, 02 Feb 2026 23:35:52 GMT

Bash commands can be used in a script to perform complex functionsENDOFTEXT The keyword ENDOFTEXT can be used to denote a multiline string Standard Out and Standard In acts as buffers to for input and output If statements can be implemented via the syntax if ( [condition] ); then # function fi Integer comparisons are represented by -ne(!=), -lt(<), -gt(>), -le(<=), and -ge(>=)Mathematical expressions can be represented by (( [expression] ))Standard integer operations such as ++, // can be usedConstants can be denoted via the readonly modifierFunctions can be implemented via the syntax afunction() { # function } Arguments can be given by $1, $2, and so on.For advanced parsing of command line arguments, getopts can be used while getopts ":s" option; do case "$option" in s) size_flag=1 ;; ?) printf "Error: Unknown option '-%s'.\n" "$OPTARG" >&2 exit ;; esac done This will check for the command line argument -s, the : in front of the s suppresses all built-in errors, and since there is no : after the s, there is should be no value afterUpon fail this will output to Standard Out and output with an error, the "$OPTARG" is the variable of the flag and 2 is the standard error flagShifting Arguments can be shifted via shift [number] shift "$((OPTIND-1))" Shift up until the first available argument (removes all flags)

Resistor Example_3

Mon, 02 Feb 2026 21:53:21 GMT

Resistor Example_2

Mon, 02 Feb 2026 21:40:19 GMT

Resistor Example_1

Mon, 02 Feb 2026 21:36:02 GMT

Resistor Example_0

Mon, 02 Feb 2026 21:29:28 GMT

Resistor Example

Mon, 02 Feb 2026 21:28:45 GMT

Exact Equation Example_0

Mon, 02 Feb 2026 00:36:21 GMT

Separable Equation Example

Mon, 02 Feb 2026 00:35:25 GMT

Exact Equation Example

Tue, 27 Jan 2026 19:06:51 GMT

Electric Field of Surface Charge Example

Tue, 27 Jan 2026 18:06:28 GMT

Electric Field of Surface Charge

Tue, 27 Jan 2026 17:10:08 GMT

Electric Field of Line of Charge

Tue, 27 Jan 2026 15:56:43 GMT

Consider a uniformly distributed line charge

Line of Charge Example

Tue, 27 Jan 2026 15:55:36 GMT

Electric Field of Continuous Charge Example

Tue, 27 Jan 2026 15:33:00 GMT

Electric Field of Continuous Charge Example_0

Tue, 27 Jan 2026 15:32:53 GMT

Method of Integrating Factor Example

Tue, 27 Jan 2026 15:13:46 GMT

Method of Integrating Factor General Method

Fri, 23 Jan 2026 01:23:59 GMT

Method of Integrating Factor

Fri, 23 Jan 2026 01:02:02 GMT

For non integrating First Order Linear equations an integrating factor can be used to convert the equation into a Type I equation, the factor must also be solved then substituted before integrating to reach the final answer

Method of Integrating Factor General Method_0

Thu, 22 Jan 2026 18:58:52 GMT

Method of Integrating Factor Example_0

Thu, 22 Jan 2026 18:47:47 GMT

First Order Linear Example

Thu, 22 Jan 2026 18:23:24 GMT

SI Unit

Wed, 21 Jan 2026 22:22:16 GMT

The standard units for physics is the International Standard of UnitsThe system is built on 7 base units defined using the fundamental constants c, h, and the vibration of a Caesium-133 atom

Bond Length

Wed, 21 Jan 2026 22:14:09 GMT

Bond length is the average distance between the nuclei of two bonded atoms in a moleculeBond lengths are usually measured in picometers (pm) or angstroms (Å) using experiment techniques such as x-ray diffraction or Molecular Spectroscopy
Bond length usually decrease when Bond Order increase or when Bond Energy increase

Ordinary Differential Equation Example

Tue, 20 Jan 2026 18:39:58 GMT

Scalar Field

Tue, 16 Dec 2025 20:43:17 GMT

A scalar field is a function of several variables of the typeWhich takes in a multi-dimensional vector and outputs a scalar With the domainAnd the imageThe level set for a scalar field is the set of all input values that output The level set of is a circle Check continuity (plug it in) Check a few paths Any two paths differ then the limit DNE The limit can not be proved to exist via exhaustion Try Polar Coordinates Solving limits via polar cordinate

Valance Bond Theory

Tue, 16 Dec 2025 20:01:35 GMT

The valance bond theory is a quantum mechanical method of describing Covalent Bonding, where a bond is created through the atomic overlap to two atomsA constructive overlap of two orbitalsA single bond is a sigma bond A double bond is a sigma bond and a pi bond A triple bond is a sigma bond and two pi bondsValance bond theory so far, can not account for polyatomic molecules such as To account for this, carbon goes through hybridization,One core electron is promotedThen carbon goes through hybridization, creating 4 equivalent hybrids The superscripted indicates that p orbitals have combined with one s orbital
Though valance bond theory is in some ways better than using Lewis Structures, there are still some problems with the theory No indication of what hybrid to use Valance bond theory still require resonance
Molecular Orbital Theory was made to correct these

Line Integral Example

Tue, 16 Dec 2025 17:41:50 GMT

Flux

Tue, 16 Dec 2025 17:39:20 GMT

A flux integral of a vector field through a surface with Orientation can be shown asThis integrated the flow of a vector field through a surface at an orientationTo parameterize the integral, note that Since can be parameterized to

Chain Rule

Tue, 16 Dec 2025 17:37:56 GMT

The chain rule represents the behavior of derivatives in chained functions Chain Rule in 1D The chain rule can be generalized to dimensionsAssume has continuous derivatives of all order. Expressin terms of the Partial Derivative of Relationships that are not dependent on a single variableEvery implicit relation among variables and can be expressed as a level set

Application of Integrals

Tue, 16 Dec 2025 17:37:35 GMT

Density can be determined as a ratio of mass over volume, this ratio can vary continuously as a function of locationIn situations where density of an object is not uniform, integrals can be used to determine center of massWhere is the position of the point In non-uniform objects, the moment of inertia measures how spread apart the mass is from the center of mass, this can determine how hard it is to spin the objectGiven a solid ball, a hollowed out ball, and a hoop, all with the same mass, find the smallest moment of inertia Moment of Inertia of a Solid Ball Substitute

Angular Momentum

Sun, 14 Dec 2025 20:04:00 GMT

The change in angular momentum is defined as cross product between the radius and momentum The change in angular momentum is also the Torque Consider a rotating figure stature with their arms stretched, if they pull in their arms the angular momentum must be conserved
Since the moment of inertia decreases as the arms pull in, the angular momentum must increase, therefore the skater spins faster
However, consider the Kinetic EnergySince is constant and decreases, the kinetic energy must have increased, which leaves the question: what's doing the work?
When the figure skater does positive work to pull their arms against the centrifugal force, this internal force is capable of changing the system's kinetic energy

Precession Example

Sun, 14 Dec 2025 19:54:39 GMT

Precession

Sun, 14 Dec 2025 19:50:02 GMT

Precession is the change in orientation of the rotational axis as a response to external torque The angular speed of precession can be givenConsider a rapidly spinning bicycle wheel with on end of its axle supported on a pivot
Because the wheel is rapidly spinning, the Angular Momentum is in the direction of the axle
Gravity will pull down the wheel at the center of mass therefore the Torque must be perpendicular to the direction of the angular momentumThe torque will apply a small change in angular momentum but because of the already existing momentum, only the direction of the momentum will be changed slightly cause the wheel to precess around the pivot

Torque

Sun, 14 Dec 2025 19:25:44 GMT

Torque is the twisting force that causes angular acceleration While torque is a vector, the direction of the torque changes constantly so it is often not expressed as vectorIn 2D the change in work is defined asHowever, for rotations, this is defined asTorque in a direction depends on two perpendicular components, torque in the direction will depend on force in the and directionsMore simply, torque can also be defined as the change in Angular Momentum in the same directionThe torque can also be defined as the projection of onto the perpendicular to Lever Arm The perpendicular vector to is called the lever arm. : Lever arm

Angular Momentum Figure Skater

Sun, 14 Dec 2025 19:14:10 GMT

Minimum Spanning Tree Dijkstra's Example

Sun, 14 Dec 2025 01:49:22 GMT

Car on a Circular Track

Sat, 13 Dec 2025 01:28:34 GMT

Consider a car on a flat circular trackThe three forces acting on the car are, gravity , the normal force , and the static friction of the track We can separate this into two equations, one for the vertical direction and one for the radial direction And since for the car to turn without slipping the static friction must be not exceed the maximum static friction forceNow consider the same car on a banked slope, the banking allows the car to turn at much higher speedsSimilarly, the three forces acting on the car are, gravity , the normal force , and the static friction of the track but the normal and frictional forces are slopedThe equations are motion can be given below

Car on a Circular Track Banked Sloped Setup

Sat, 13 Dec 2025 01:21:21 GMT

Car on a Circular Track Setup

Sat, 13 Dec 2025 01:13:18 GMT

Green's Theorem

Fri, 12 Dec 2025 01:27:42 GMT

Green's theorem is a way to find the line integral of a continuous path over and a conservative Vector FieldWhere is a (simple) region on the plane And is a counterclockwise-oriented, simple, closed curve, forming the boundary of D. And which is a continuously differentiable vector field Green's Theorem Example_0.excalidraw

Ligand

Fri, 12 Dec 2025 00:50:57 GMT

Ligands are simple molecules or ions that combine with metals to make metal ionsIn this coordinated ion, both and are ligandsNH₂CoClClNH₂H₂N

Crystal Field Theory

Fri, 12 Dec 2025 00:39:02 GMT

Crystal field theory is based on how how atomic orbitals interact with the surrounding ligands

Transition Metals

Fri, 12 Dec 2025 00:28:07 GMT

Transitions metals are elements from Group 3 – 11 and are called d-block metals, it's important to note that and are not transition metalsTransition metals usually: Exhibit variable oxidation (valence) states. Form compounds that have incomplete d shells.
Form compounds that are paramagnetic (attracted to magnets) Form colored compounds. Form compounds that consist of complex ions Exhibit catalytic properties as the metal or in compounds

Rotation of Molecules

Thu, 11 Dec 2025 23:40:41 GMT

Molecules are considered to be Rigid Bodies where Rotation occurs about the center of Gravity
In order to excite a rotational transition, the molecule must have a permanent Dipole Moment, this is because light interacts with the electric dipole of the molecule, if there is no permanent dipole, light can not excite the moleculeIn addition, absorption of light is only possible for transitions that obey the selection rule Consider a microwave and a cup of waterBecause the water is polar, microwave radiation excites molecular rotations in the water, the rotation of the water hits other water molecules, causing them to translate and transfer the microwave radiation into heatThe cup however, is most likely not polar and therefore the molecule rotation has no effect on the cup

Rigid Body Dyanmics

Thu, 11 Dec 2025 23:14:31 GMT

We can define force as the rate of doing WorkForce can also be defined as the rate of change of momentumUsing the work definition, when separating variables for Using the momentum definition

Heisenberg Uncertainty Principle

Thu, 11 Dec 2025 23:13:09 GMT

For particles that exhibit Wave behavior, the more we know about the position, the less we know about the trajectory, this is true vice versa

Vibration of Molecules

Thu, 11 Dec 2025 23:12:52 GMT

While atoms can only exhibit translation, molecules can exhibit translation, rotation, and vibrationThe ability of a molecule to vibrate depends on the energy required to increase or decrease internuclear separation from the equilibrium valueThis behavior can be model by using springs and Hooke's LawIn classical mechanics, the oscillation frequency is given byWhere is the force constant And is the reduced mass ()
In Quantum Mechanics however, each level of energy is quantizedBecause the spacing between energy level is Independent of the electron, the change in energy level is restricted to
Some vibrations do not change the Dipole Moment of the molecule, for example, a stretching movement does not typically induce a change in the dipole moment, these vibrations are called IR inactiveFor a molecule to be IR active, the molecule must exhibit a change in the dipole momentFor polyatomic molecules, the number of vibration modes can be given belowConsider water (), using the non-linear vibrational equationWe end up with vibrational modes

Vibration of Molecules Water Vibration

Thu, 11 Dec 2025 22:58:17 GMT

Electronic Spectroscopy

Thu, 11 Dec 2025 22:37:48 GMT

Electronic Spectroscopy works by transferring an electron from a Molecular Orbital to another orbital

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Thu, 11 Dec 2025 22:29:43 GMT

Molecular Spectroscopy

Thu, 11 Dec 2025 22:09:27 GMT

Because molecules are three dimensional, the energy of molecules also depend on rational and vibrational propertiesThe energy of a molecule can be represented by“Pasted image 20251211170925.png” could not be found.Absorption spectra for molecules are typically recorded by spectrophotometersIn an experiment the intensity of the light transmitted through the sample cell is compared to a reference cellSpectra can be presented in either transmittance or absorbance modes, 0% transmittance means no light is observedTransmittance can be defined by Absorbance can be defined by Absorbance can also be given generally byWhere is the absorption coefficient And is concentration And is the molar extinction coefficient And is the cell length

Molecular Orbital Theory

Thu, 11 Dec 2025 21:59:30 GMT

Molecular orbital theory is created to fix certain problems with Valance Bond Theory and VSEPR Theory Molecular orbitals are derived from a superposition of the atomic orbitals, the theory relies on the fact that as two orbitals overlap, they would interfere constructively or destructivelyElectrons would bond in pairs to either the bonding orbital or the antibonding orbital depending on the lowest energyThe number of molecular orbitals equal the sum of atomic orbitals The number of electrons in molecular orbitals equals the sum of the number of electrons in the atomic orbitalsA bonding orbital is the sum of two orbitals, and usually has lower energy than the orbital of any one atom
Consider two hydrogen atoms, as the atoms approach each other the orbitals will overlap and interfere constructively or destructively based on the phaseNot to be confused with antibonding orbitals, nonbonding orbitals are orbitals with inappropriate symmetry are non-bonding because the constructive and destructive interference cancel each other out Molecular Orbital Theory Nonbonding Example.excalidrawUnlike essentially nonbonding orbitals, rigorously nonbonding orbitals are unable to interact at all Identify orbitals from valance shell Combine pairs of atomic orbitals Atomic Orbitals combine most effectively with other atomic orbitals of the same or similar energy The effectiveness of their combination is their overlap Adjust for charge Accommodate electrons in order of increasing energy When molecular orbits have the same energy, electrons enter orbitals with the same spin before pairing the spins The orbitals from combining two atoms will produce 8 molecular orbitals The bond order of a molecular orbital can be given by
An increase in bond order generally increase Bond Energy and decrease Bond LengthTo predict the molecular structure using Molecular Orbital Theory, calculate all possible structures for a molecule then chose the structure with the lowest energyConsider the Molecular Orbital Diagram for fluorine ()Molecular Orbital Theory Diatonic Example.excalidrawThe 8 atomic orbital from each atom gives 8 molecular orbitals of which only 7 are filled, fluorine has 7 valance electrons which means only 14 electrons are filled in the molecular orbitalsThe final electron configuration of is Since the bond order can be given by the number of bonding orbitals minus the nonbonding orbitalsIn molecular orbital theory, a molecule is paramagnetic is there are unpaired electrons in any of the orbitals
For example, because fluorine doesn't contain any unpaired electrons it is diamagnetic
However, oxygen contains two electrons in the orbit, making it paramagnetic

Atomic Orbital

Thu, 11 Dec 2025 21:52:39 GMT

Atomic orbitals are complex mathematical functions that can be separated into two components: a radial wavefunction and an angular wavefunction While the energy is described by a single Quantum Number , the wavefunction is described by three quantum numbers , ,

Bond Energy

Thu, 11 Dec 2025 21:20:38 GMT

The average bond energies within a reaction can be used to predict if a reaction will be favorable or notEnergy is released in the formation of a bond meanwhile it is absorbed in the breakage of a bondEnergy change in the formation of one mole of a compound from its components Lattice energy only exists for ionic compounds and can be given byThe energy required to isolate a molecule is determined by it's energyAtoms are more stable in molecules since there are less valance electronsConsiderBreaking apart and requires energy to be spent respective to their bond energy Creating releases its bond energyThis equation will release overallFor similar systems, bond energy increase with increase in Bond Order while bond length decreases with increase in bond order

Molecular Orbital Theory Diatonic Example

Thu, 11 Dec 2025 19:57:35 GMT

Molecular Orbital Theory Nonbonding Example

Thu, 11 Dec 2025 19:43:51 GMT

Molecular Orbital Theory Example

Thu, 11 Dec 2025 19:33:00 GMT

Variation of Orbital Energies

Tue, 09 Dec 2025 23:54:17 GMT

In Valance Bond Theory, as the atomic number increases, 3s, 3p drops more rapidly because they penetrate the core more effectivelyHowever the 3d orbital remains unchanged up until around (Z = 19)This causes orbital energy to be more sensitive to the atomic number as the order decreases
Unusual configurations - - - - - There is a drop in the first ionization energy from to , this goes against the trend of ionization energy increasing across a period
The reason why lies within the Electronic Configurations of the two elements: has a configuration of , which is creates a half-filled shell making N more stable than 's configuration of which undergoes more electron propulsion

Electron Configuration

Tue, 09 Dec 2025 23:44:38 GMT

Electron configuration is the arrangement of electrons within the orbitalEach electron can be characterized by a set of four Quantum Numbers which defines the the electron Orbital's shape, orientation, and spin
Electron configurations are prone to follow principles such as the Aufbau's Principle, Hund's Rule, and the Pauli Exclusion PrincipleWhile most transition metals have the configuration , and have configurations of and respectively; this can be explain by the fact the half-filled shells are more stable because electrons inhibit parallel spin

Hund's Rule

Tue, 09 Dec 2025 23:32:17 GMT

If there are more than one Orbital in a subshell the electrons are added to different orbitals with parallel spins rather than pairing two electrons in the same orbital

Aufbau's Building Up Principle

Tue, 09 Dec 2025 23:32:17 GMT

The Electron Configuration of atoms is made by filling orbitals with electrons in order of increasing energy.The sequence of orbitals can be remembered using the *n + l rule The energy increases with the n + l value. For two subshells with the same value of n + l the lower value n is completed first The order is as follows 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d

Electron Spin

Tue, 09 Dec 2025 23:30:06 GMT

The spin of an electron can be considered the fourth quantum numberMultiple electrons in a system are hard to evaluate since each body affects the other electron Because all electrons affect each other in an orbit, the effective charge of an electron can be given minus the shielding constant Core penetration in Effective Nuclear Charge While 3s and 3p have higher average distances than 3d, they have local maximums within the core area, because of this, their effective nuclear charge is higher If a p orbital has a node () at the nucleus, how do electrons get from one side to the other?While in classical mechanics, the particles must move to via continuous translation, in Quantum Mechanics this is not the case, the expression is only true when the electron is forced to behave as a particle

Euler Path

Sun, 07 Dec 2025 18:45:52 GMT

An Euler path is a path that visits every edge exactly onceAn Euler circuit is a Euler path that starts and ends on the same node

Spanning Trees

Fri, 05 Dec 2025 02:18:20 GMT

The depth first spanning tree is a search tree used by Depth First Search Add a tree edge for every graph edge taken by DFS Add a back edge for every skipped edgeIf the root of the DFS spanning tree has two outgoing edges then the root is an articulation point

Graph

Fri, 05 Dec 2025 02:16:55 GMT

Graphs An undirected graph is connected if there is a path for every vertex to every other vertexA graph that inhibits biconnectivity can not have any edge removed without losing connectivity of the graphAn adjacency matrix maps the relation between any two points on a graphGraph Example_0.excalidrawWhile an adjacency matrix plots all relations, the adjacency list is more space efficient by removing

Minimum Spanning Tree Example Union-Find

Tue, 02 Dec 2025 19:26:34 GMT

Minimum Spanning Tree Kruskal

Tue, 02 Dec 2025 19:04:11 GMT

Sorting

Tue, 02 Dec 2025 03:55:30 GMT

Sorting algorithms aim to take a disorganized array and arranges it into a sorted arrayBubble sort iterates over the Array, swapping value pairs that are not sorted, is then set to and the process repeats until is 0. is the length of the unsorted array is the highest index where a swap occursBest: (Given the best possible array) Worst: (Given the worst possible array) Average: (Given a random array) Overall: Basic Implementationpublic static void bubbleSort(int[] array) { int len = array.length; while (len > 0) { int n = 0; for (int i = 1; i < len; ++i) { if (array[i - 1] > array[i]) { swap(array, i - 1, i); n = i; } } len = n; } }
Selection sort iterates over the Array and repeatedly finds the smallest element in the unsorted array, then swaps it to it's resting index
Reverse Selection Sort Selection Sort can also move the maximum value to its resting index and still have the same Time Complexity Best: (Given the best possible array) Worst: (Given the worst possible array) Average: (Given a random array) Overall: Practical Time Even though bubble sort have better time complexity on paper, selection sort swaps less than bubble sort, equating to better practical speed Basic Implementationpublic static void selectionSort(int[] array) { for (int i = 0, i_bound = array.length - 1; i < i_bound; ++i) { int min_index = i, min = array[i]; for (int j = i + 1; j < array.length; ++j) { if (array[j] < min) { min_index = j; min = array[j]; } } if (min_index != i) { swap(array, i, min_index); } } } Insertion sort creates a subarray with the first elements in order in respect to other elementsBest: (Given the best possible array) Worst: (Given the worst possible array) Average: (Given a random array) Overall: Basic Implementationpublic static void insertionSort(int[] array) { for (int i = 1; i < array.length; ++i) { int k, current = array[i]; for (k = i - 1; k >= 0 && array[k] > current; --k) { array[k + 1] = array[k]; } array[k + 1] = current; } }
Instead of swapping every pair, like in Bubble Sort, Odd-Even Transposition considers all odd pairs using Parallelization, swapping when necessary, considers all evens When no swaps are made after both an odd and even passthrough, the algorithm endsParallelization Since Java can not support parallelization on the CPU, the Odd-Even Transposition sort can not be implemented natively in Java To use heap sort convert the array to a max heap then perform deleteMax() operations to get the array in reverse orderBest: (Given the best possible array) Worst: (Given the worst possible array) Average: (Given a random array) Overall: (Distinct elements )private static void swap(int[] array, int i, int j) { int temp = array[i]; array[i] = array[j]; array[j] = temp; } private static void percolateDown(int[] array, int i, int n) { int child = 2 * i + 1, temp; for (temp = array[i]; child < n; i = child, child = 2 * i + 1) { if (child != n - 1 && array[child] < array[child + 1]) { child++; } if (temp < array[child]) { array[i] = array[child]; } else { break; } } array[i] = temp; } public static void heapsort(int[] array) { for (int i = array.length / 2 - 1; i >= 0; i--) { percolateDown(array, i, array.length); } for (int i = array.length - 1; i > 0; i--) { swap(array, 0, i); percolateDown(array, 0, i); } } Merge sort relies on two sorted arrays being able to combine into a single sorted array, merge sort divides the array into two arrays then recursively sorts the arraysBest: (Given the best possible array) Worst: (Given the worst possible array) Average: (Given a random array) Overall: private static void mergesortHelper(int[] array, int[] scratch, int low, int high) { if (low < high) { int mid = low + (high - low) / 2; mergesortHelper(array, scratch, low, mid); mergesortHelper(array, scratch, mid + 1, high); int i = low, j = mid + 1; for (int k = low; k <= high; k++) { if (i <= mid && (j > high || array[i] <= array[j])) { scratch[k] = array[i++]; } else { scratch[k] = array[j++]; } } for (int k = low; k <= high; k++) { array[k] = scratch[k]; } } } public static void mergesort(int[] array) { int[] scratch = new int[array.length]; mergesortHelper(array, scratch, 0, array.length - 1); } Quick sort partitions Counting sort works by counting all appearances of a number within an array then rebuilds an array in orderBest: (Given the best possible array) Worst: (Given the worst possible array) Average: (Given a random array) Overall: Space: Topological Sort works by computing the indegree of all vertices and putting all indegrees of 0 in a min-heap, then it repeatedly extracts the lowest number indegree 0 then updates adjacent vertices indegree: amount of adjacent vertices that point towards the given vertex
O

Uniform Circular Motion

Sun, 30 Nov 2025 17:46:20 GMT

Uniform Circular Motion occurs when an object moves in a circular path at a constant speed, even though the speed remains the same, the direction of motion changesAn object moving in a circle of radius with angular velocity has a position vectorThe linear velocity is always tangential to the circle:The magnitude of velocity can be expressed asAlthough the speed remains constant, the direction of changes, producing an inward (centripetal) acceleration:or equivalently,This acceleration is responsible for changing the direction of motionUsing Newton’s second law,The net force required to maintain circular motion is thus inward (centripetal):This is known as the centripetal force, which is not a new type of force but the result of other forces The time taken to complete one revolution (the period) is given by:The frequency (number of revolutions per second) is:

Conservative Force

Sun, 30 Nov 2025 17:41:33 GMT

We can define work as the change in energy over a distanceBy equating What..Take the curl of

Newton's Law

Sun, 30 Nov 2025 17:41:15 GMT

Newton's Laws of motion describe the relationship between an object's motion and forces that act upon itThe first law states that an object in motion stays within that motion unless acted upon by an external forceThe second law states that the Force of an object is equal the object's mass times its acceleration.Newton also defines the force as the change in momentum at a given timeOne of the fundamental forces is the gravitational force Where is the direction from one body to the otherOn earthNewton's third law states that if two objects exert a force on each other, those forces will have the same magnitude but opposite directions

Friction on Inclined Planes

Sun, 30 Nov 2025 17:35:24 GMT

The normal force of an incline plane is given by The force in the direction down the slope is given by Static friction matches up until which is given by The critical point of where static friction fails to stop the object can be given byFriction on Inclined Planes Pulley.excalidraw

Time Derivative

Tue, 25 Nov 2025 15:54:01 GMT

The derivative of a vector in respect to time will always be perpendicular to the vectorProduct rule for vectors Given Then To rotate a vector to its time derivative multiply by a vector orthogonal to both vectors Where is orthogonal to both and

Orientation

Thu, 20 Nov 2025 23:36:27 GMT

A surface is oriented if there is a consistent choice of normal vectors This can also be achieved by parameterization of of the surfaceTo flip an orientation, negate a component, the surface will be flipped in respected to the flipped component

Show Flipped

Minimum Spanning Tree Example

Thu, 20 Nov 2025 18:30:59 GMT

Sorting Example

Thu, 20 Nov 2025 17:45:58 GMT

General Roation in 3D Example

Thu, 20 Nov 2025 15:44:43 GMT

Pauli Exclusion Principle

Wed, 19 Nov 2025 03:52:34 GMT

No two numbers can have the same set of four Quantum Numbers
This is the basis for opposite Electron Spin, an orbital can only hold up to 2 electrons which can not be the same spin

Quantum Number

Tue, 18 Nov 2025 23:36:14 GMT

Elements can be defined using a set of quantum numbers that are solutions to Schrödinger's Equation.
These are used in Valance Bond Theory to predict shape.
Quantum numbers can be used to find the Shape of Atomic Orbitals Given by the symbol Determines the overall size and energy scale Number of orbitals in shell Maximum electrons in shell Given by the symbol
Determines the orbital angular momentum and the shape of the Orbital Number of orbitals in a subshell = Maximum electrons in a subshell = S orbitals Even though S orbitals have 0 angular momentum, the electron still has motion, the electron can be imagined as having a closed linear path Given by the symbol Specifies the orientation of the orbital along an axis Values are integer steps; there are possible values for a given For example can have values of
Main Page: Electron Spin Given by the symbol Intrinsic angular momentum of the electron Two possible spin states per orbital Shown experimentally by the Stern–Gerlach experiment Splitting beams in an inhomogeneous magnetic field

Covalent Bonding

Tue, 18 Nov 2025 22:15:28 GMT

Atoms in the molecule are held together by sharing electronsCovalent bonds are often represented by a line (–) between two atomsCH₃OHOOH

Atomic Orbital Solutions

Tue, 18 Nov 2025 22:07:13 GMT

Shapes of Orbitals

Tue, 18 Nov 2025 21:13:13 GMT

Orbitals have differing shapes based on interactions between electrons in the Valance Bond Theory
Every electron has a wave function that determines the probability that the electron is at a certain radiusThe probability density can be determined by squaring the wave function Nodes are spaces in the wave function with a zero probability, meaning no electrons will occupy that spaceBecause the wavefunction goes to infinity, the size of an atom can not be fully defined A convention is to use to boundary where there is a 90% probability of the electron can be locatedWhile represents the probability at a point, represents the probability at a certain radius
Where is the radial wavefunctionIn a radial distribution function, spaces with zero probabilities are radial nodes, which appear as spheres without any electron occupation

Atomic Radii

Tue, 18 Nov 2025 21:11:26 GMT

In Bohr's Theory, the radius of an Element can be given byWhere is the Bohr radiusAtomic radius generally increases down a group and decreases across a period
Quantum Representation of Radii The wavefunction of an atom can be used to more accurately describe the radius of atoms

Wave

Tue, 18 Nov 2025 20:42:57 GMT

Forces like electromagnetic radiation and light behave similar to wavesWave-Particle Duality Electromagnetic radiation and light in reality exhibit Wave-Particle Duality and act like both waves and particles Waves that combine are in phase with each other interfere constructively, increasing the amplitude of the resulting wave, waves that interfere destructively result in a lower amplitude wayTraveling waves are waves that change travel along a distance, standing waves change in place A wave traveling along length of rope will exhibit standing wave behavior while a guitar string will exhibit standing wave behavior

Wave-Particle Duality

Tue, 18 Nov 2025 20:42:53 GMT

Small particles, at times can display Wave-like properties.
Electrons in a Bohr orbit may be associated with a circular standing wave wave, interfere is the reason why only certain wavelengths appear in each orbit
Einstein's equation can be linked to Planck's equation to form a relation between the wavelength and momentum of a photonAnd since is the speed of a photonAnd since
Heisenberg Uncertainty Principle Because of the Heisenberg Uncertainty Principle, the more known about a particle's position, the less known about its momentum and vice versa

Dipole Moment

Tue, 18 Nov 2025 20:37:24 GMT

If a shape of a molecule is known, we can determine the dipole moment of a molecule by combining the dipole moment of its individual bondsIn , a flat shape, the dipole moments of the bonds cancel outIn , a bent shape, the dipole moments of the bonds add together to be polar

Quantum Mechanics and Atomic Theory

Tue, 18 Nov 2025 20:23:34 GMT

Rutherford proposed a planetary model for the atom where most of the mass is concentrated in the nucleus while electrons "orbit" around the nucleusDeath Spiral of an electron A common critique of Rutherford's model is that an electron can not keep orbit forever, as the the electron orbits the electron loses energy, eventually spinning into the nucleus A central idea of quantum theory is that energy isn't continuous but comes in discrete packetsOriginally light was thought of as a particle, however Maxwell suggested that light consists can behave like a Wave.Amplitude Wavelength (): Frequency (): Speed ()Speed of a waveFrequency of a waveLight is a form of electromagnetic radiation, the electric component influences charged particles, the magnetic field influences moving charged particlesHumans can be seen in the dark by detecting heat, in a similar way, all objects emit radiation (infrared light) that can be detectedObjects that emit blackbody radiation emit energy dependent on the temperature of the object Incandescence: Light due to temperature Chemiluminescence: Light due to a chemical reaction Fluorescence: Light emitted by an object that absorbed light

Rotation

Tue, 18 Nov 2025 15:31:56 GMT

Given a system of particles with no external forcesWhere is the center of massRotation does not induce any external forcesIn a rigid body, the distance between the fixed center of mass and a point will be constant.Conversion from polar to cartesianThe change in position for rotationThe position in rotation for a given can be given as follows

General Time derivative

Fri, 14 Nov 2025 03:38:55 GMT

Given The general formula for a time derivative of a position vector can be defined asWhere is the time derivative of the angle Construct a unit vector to by using cross product

Green's Theorem Example_0

Fri, 14 Nov 2025 03:37:22 GMT

Line Integral

Thu, 13 Nov 2025 23:40:41 GMT

Also called a path integral, is an integral with the domain of , the image of a curve There are several version of line integrals with differing usesLine integrals in respect to arc length measures the area between the curve and the Scalar Field the integrated shape is a extruded path
In order to integrate the line integral, convert in terms of using Reparameterization
Line integrals in respect to a coordinate measures the area between the linear change in the curve and the Scalar FieldGiven , find the line integral in respect to , , and arc length

Line integrals in respect to the Vector Field measures the alignment between the curve by integrating the dot productIf and is a continuous path from to thenSolution to

On a closed curve, a line integral conservative vector field will always equal

Green's Theorem Example

Thu, 13 Nov 2025 23:07:36 GMT

Green's Theorem Example Not Simple Not Closed

Thu, 13 Nov 2025 22:53:29 GMT

Green's Theorem Example Simple Not Closed

Thu, 13 Nov 2025 22:53:12 GMT

Green's Theorem Example Not Simple Closed

Thu, 13 Nov 2025 22:52:42 GMT

Green's Theorem Example Simple Closed

Thu, 13 Nov 2025 22:52:25 GMT

Benzene

Thu, 13 Nov 2025 16:29:47 GMT

Types of Forces

Thu, 13 Nov 2025 15:57:02 GMT

Forces can be classified into different types depending on Gravitational Force $$ \vec{F}=-\frac{GmM}{r^2}\hat{r} $$ Electric Force $$ \vec{F}=\frac{1}{4\pi\epsilon} \frac{qQ}{r^2}\hat{r} $$Phenomenological motion is constraining motion that results from complex interactions between objects the force is observed and found experimentallyStatic Friction: Negates motion parallel to a surface Kinetic Friction: Opposes motion parallel to a surface Normal: Opposes motion normal to a surface Tension: Constraint of a string, force in direction in the direction of a string at its length Spring: Constraints of a spring, force in direction opposite to the resting position

Power

Thu, 13 Nov 2025 15:56:46 GMT

Power is the change in work over time or how fast the work is being doneSincePower can be defined asPower can be generalized in 2D by using vectors

Rotation Example

Thu, 13 Nov 2025 15:33:06 GMT

Tree Searching

Wed, 12 Nov 2025 23:26:00 GMT

Searches along a length firstSearches along the row first

Friction on Inclined Planes Pulley

Wed, 12 Nov 2025 23:24:45 GMT

Friction on Inclined Planes Walking

Wed, 12 Nov 2025 23:24:38 GMT

Friction on Inclined Planes Example

Wed, 12 Nov 2025 23:24:29 GMT

Vector Field

Wed, 12 Nov 2025 23:22:10 GMT

A vector field is a function where every vector input will output a vectorA vector field is conservative if there is a Scalar Field (potential) that satisfies
Conservative vector fields can be called gradient vector fields because a conservative vector field is a Gradient of a scalar fieldA vector field can be proven to be not conservative by using proof by contradiction
Non Conservative Vectors There are exceptions to this proof since there are non conservative vector fields that satisfies A potential can be found by defining as and using the equality

Graph Example_0

Wed, 12 Nov 2025 23:16:12 GMT

Tower of Hanoi

Wed, 12 Nov 2025 23:03:34 GMT

The tower of Hanoi is a proposed game where rings are placed in descending size onto a peg and the goal of the game is to move those rings onto a different peg, all rings must be in descending order at all times

Binary Tree

Wed, 12 Nov 2025 23:03:28 GMT

In Binary Trees, the number of children is at most twoThe amount of nodes within a Binary Tree is exponential and can be given by where is the total levels in the tree Binary trees are considered full if every node is a leaf or has two children Binary trees are considered complete if all levels (except possibly the last) is completely filled and every node is as far left as possibleIn Binary Search Tree the root is the key, all left children are less than the key while all right children are greater than the key

Graph Example

Wed, 12 Nov 2025 23:02:57 GMT

Time Complexity

Wed, 12 Nov 2025 22:58:38 GMT

The time complexity of a function can be solved by using a 5 step processAims to reduce the problem to a single constant If base case, return directly. Reduce input to a smaller instance. Recursively solve the smaller instance. Extend/adapt that solution to solve the original. Find the time complexity of given Time Complexity Example B.excalidraw]

Shapes of Molecules Using VSEPR

Wed, 12 Nov 2025 15:31:10 GMT

Lewis Structure molecules does not provide information about the shape of the molecule VSPR theory can be used to determine the shapeVSEPR, which stand for Valance Shell Electron Pair Repulsion, can be used to estimate a shape of a molecule. VSEPR is not in itself a theory but can be used as an aide Molecules will minimize electron pair repulsion which can determine its shapeSteric numbers can be found by adding the number of coordinated electrons with the number of lone pairs By using the steric number the shape of a molecule can be found2: Linear 3: Trigonal Planar 4: Tetrahedral 5: Trigonal bipyramidal 6: Octahedral 7: Pentagonal bipyramidalThe preferred rotation in any of these shapes minimizes interactions of lone pairs first then the interactions of line pairsShapes can be drawn in any rotation as long as the lone pairs minimizes interactionsBecause electrons bonded between two atoms take up less space, they are distributed more evenly, for example the Hydrogens in are evenly spaced Electrons in lone pairs take up more space and decreases the angle of other bonds, the Hydrogens in are closer to each other than

Line Integral Fundamental Theorem

Thu, 06 Nov 2025 23:51:06 GMT

Vector Field Conservative Vector Proof

Thu, 06 Nov 2025 23:29:15 GMT

Polar Vector

Thu, 06 Nov 2025 23:26:03 GMT

The variables and can be used to define coordinates in the polar plane The general equation for a polar vector value function is as followsThe variables and and can be used to define coordinates in the 3D polar planePolar vectors in 3D share similar dot and cross product rules to cartesian vectors in 3DOmega can be defined as the change in angle in the z directionSo thatThe equation for circular motion can be given by

Polar Coordinates

Thu, 06 Nov 2025 23:25:46 GMT

The magnitude of a vector is given by And its corresponding angle given by Such that And its time derivative Rotate a Polar Vector to its time derivative is similar its Cartesian equivalentThe period and frequency of an polar vector are as followsA Polar vector also be defined asAngular Direction The angular direction is by definition the perpendicular direction of

Cylindrical Coordinates

Thu, 06 Nov 2025 23:25:00 GMT

Volumes can be expressed in terms of Polar Coordinates with the terms The conversion for cylindrical coordinates are the same as polar coordinates in 2D

Spherical Coordinates Volume

Thu, 06 Nov 2025 22:19:43 GMT

Parallelization

Wed, 05 Nov 2025 23:44:12 GMT

Parallelization is the process of running multiple processes at the same time

Directional Derivative

Wed, 05 Nov 2025 23:39:10 GMT

Let be a unit vector, the directional derivative of can be given byFind the direction derivative of the function at the point in the direction of Directional Derivatives Example.excalidrawTake the unit vector of Take the Gradient of
Dot the Gradient with

Extrema

Wed, 05 Nov 2025 23:38:27 GMT

A local minimum of a Scalar Field is a value such thatSimilarly, a local maximum of a scalar field is a value such thatWhere the domain is an open set containing
The local extrema of a Scalar Field can be determined byFor all positions A critical point of a function is a position such that is not differentiableIf is a local extremum, then is a critical point

Set

Wed, 05 Nov 2025 23:36:50 GMT

A collection of objects that does not allow duplicates unlike Arrays that do allow duplicateinsert(element): remove(element): contains(index): allAll(set): union(set): removeAll(set): intersection(set): A set with defined orders, all pairs are greater than or lesser than other pairsContains all set operations with findMin(): findMax():

Lagrange Multiplier

Wed, 05 Nov 2025 23:35:13 GMT

In most applications the control variables are constrained This can be expressed with a level set, which is known as the constraint curveA local minimum of subject to the constraint occurs at only if a is a critical point for or if the two functions are parallel for some scalar Extreme Value Theorem A continuous function on a closed and bounded set must have an absolute minimum and maximum (Extrema) On a closed and bounded set Find all critical points Find all Lagrange points The maximum is the greatest value on the combined set The minimum is the least value on the combined set

Mathematics Review

Wed, 05 Nov 2025 23:33:05 GMT

In a given Set of n. The number of subsets that can be made from elements of that set equal to n2Because the sum of i can be represented as S = 1, 2, ... , n , reversing the expansion and adding the it back onto itself can give us 2S = n+1, n+1, ... , n+1 This can be shortened as:

Searching

Wed, 05 Nov 2025 23:18:06 GMT

Probe each value in a list sequentially until target is foundBest Case: Worst Case: Basic Implementationpublic static int SequentialSearch(int target, int[] array) { for (int i = 0; i < array.length; i++) { if (array[i].compareTo(target)) { return i; } } return -1; } Sorted Only (Sorting) Probe in the middle, split into two groups, compare if the target is higher or lower, repeat search in the respective groupBest Case: Worst Base: Basic Implementationpublic static int BinarySearch(int target, int[] array) { int low = 0; int high = array.length - 1; while (low <= high) { int mid = low + (high - low) / 2; int result = target.compareTo(mid) if (result == 0) { return mid } else if (result < 0) { high = mid - 1; } else { low = mid + 1; } } return -low - 1; // Return possible index of target }

Binary Search Tree

Wed, 05 Nov 2025 23:17:42 GMT

The difference between binary search trees and Binary Tree are that all nodes to the left are less than the root while all nodes to the right are greater than the rootIt is possible to check whether a tree is a binary search tree by checking if every left node is less than and every right node is greater than
Binary Search Trees can be searched in time, similar to Binary Searchboolean checkBSTHelp(Node root, int min, int max) { if (root == null) return true; if (root.data > max || root.data < min) return false; return checkBSTHelp(root.left, min, root.data - 1) && checkBSTHelp(root.right, root.data + 1, max); } boolean checkBST(Node root) { return checkBSTHelp(root, 0, Integer.MAX_VALUE); } A binary search tree can have the potential to be very unbalanced, this can cause search times to approach rather than
This is often solved by height balanced trees, or Red-Black Tree

Lists

Wed, 05 Nov 2025 23:14:45 GMT

Lists are ordered data, it is implemented as List Interface in Java, The List Interface contains an Array Lists and Linked ListsA class that supports dynamic arrays, the size of the array grows as elements are inserted, it can store any Java objectList Instantiation Define as a List Reference instead of a ArrayList List list = new ArrayList<>(); So that the type of List can be changed easily in testing The Array List grows by every time the size of the array reaches its allotted size Array Lists do not resize when elements are removedIterator: A built-in object to loop elements in order Amortized: The expected run time over a Pre vs Post incrementer Different incrementers can be used based on necessity x++ is a post incrementer that returns the current value before modifying it ++x is a pre incrementer that modifies the value before returning it The Time Complexity of common Array List methods are as followsadd(element): : When array does not need to be expanded : When the array needs to be expanded : Amortized Time add(index, element): get(index): remove(index): indexOf(element): Linked Lists are dynamic lists made out of nodesAll objects in a Linked List must be encapsulated by a node, which contains the value and a pointer to the next node in the list They can be added and removed as needed, in any order from the ListAll Linked Lists must have a head, otherwise the linked list will be headless A headless list will not be referred and will be collected by the garbage collectorThe head refers to the starting node in the ListLinked Lists can have a tail which refers to the end of the ListJava Method Arguments When a method is called, all passed-in arguments are copies of a pointer of that value, meaning the argument is completely local int a = 8; boolean isNextEven(int number) { number++; // number will be 9; a will still be 8 return (number%2==0); }
The Time Complexity of common Array List methods are as followsadd(element): : With tail pointer : Without tail pointer add(index, element): get(index): remove(index): indexOf(element):

Heap

Wed, 05 Nov 2025 22:45:05 GMT

A heap is a complete Complete Binary Tree that is stored in Level OrderIn this way, given a node at address the address of its children and parent can be determined as such left child: right child: parent: (since integer division rounds, both children return the same parent) In Min Heaps, the root has the lowest value and all node below the root has values that are greater than or equal to the rootIn Max Heaps, the root is the highest node and all nodes are less than or equalIn order to store incomplete heaps, allocate the maximum amount of space the heap uses, leave any blank nodes as empty cellsArrays with Identical Elements Because array with identical elements does not need to be percolated, the run time for this type of array is

Hash Function

Sun, 02 Nov 2025 20:09:03 GMT

Hash functions returns a ideally unique value for every inputHash functions must follow these rules Ideally hash keys in a uniform distribution Potentially output all index values of the table Every Java object has a default implementation of hashCode The default hashCode returns a distinct integer for distinct objects but does not consider the contents of the object In order to consider the contents, the hashCode must be overridden

Probing

Sun, 02 Nov 2025 20:08:08 GMT

If the hash index of an object is already occupied, and the objects are not equal, probing can be performed to guaranteed a unique index for the functionThe next index of a cell can be given byWhere is the probing functionIn linear probing, each cell is checked linearly However, because linear probing tends to cluster around occupied cells, the function can be inefficient in near-filled tablesIn quadratic probing, the next checked cell is farther apart as increase This distributes the occupied cells more evenly and makes probing more efficientQuadratic Probing Theorem Given that is prime, then the first cells visited by quadratic probing are distinct. Therefore we can always find an empty cell if the table is at most half full. Another hashing function can be used to determine the next checked cell Where is the secondary hashing functionSimply setting a cell to null can cause problems with probing, if during search a cell is found empty, the algorithm will assume that following cells do not contain the elementInstead of setting the value to null, a cell can be lazily deleted (not removed but marked as deleted)

Line Integral Arc Length Example

Thu, 30 Oct 2025 22:28:14 GMT

Binary Tree Example

Thu, 30 Oct 2025 03:40:41 GMT

Kinematics in 1D

Thu, 30 Oct 2025 00:12:54 GMT

Given constant acceleration, the velocity at a time can be given by the following Given oscillating acceleration, the velocity at time must be integratedObjects moving thro

Time Complexity Example B

Wed, 29 Oct 2025 22:03:50 GMT

Time Complexity Example C

Wed, 29 Oct 2025 22:03:49 GMT

Orbital

Wed, 29 Oct 2025 19:45:48 GMT

Orbitals per subshell: 1 () Electron capacity: 2 electrons Shape: spherical shaped Nodes: Angular nodes = Radial nodes = Since the s orbitals penetrates the core better, s orbitals have higher effective energy than both p and d orbitals Orbitals per subshell: 3 () Electron capacity: 6 electrons Shape: dumbbell shaped Nodes: Angular nodes = Radial nodes = Similar to s orbitals, p orbitals also penetrate the core better having higher effective energy than d orbitals Orbitals per subshell: 5 () Electron capacity: 10 electrons Shapes: four cloverleaf-type or donut around a lobe Nodes: Angular nodes = Radial nodes = Orbitals per subshell: 7 () Electron capacity: 14 electrons Shapes: complex multi-lobed forms Nodes: Angular nodes = Radial nodes = f orbitals tend to be more shielded by inner electrons and less involved in simple covalent bonding compared with valence s and p orbitals. A node is a region where the wave function changes sign and becomes zero A node can either be angular or radialThe amount of angular nodes can be given by The amount of radial nodes can be given by

Formal Charge

Wed, 29 Oct 2025 19:08:01 GMT

A charge assigned to an atom within a molecule in relation to other atom which is not a real charge Its value can be determined byWhere is the total valance electrons, is the total non bonded electrons, and is the total bonded electrons

Periodic Table

Wed, 29 Oct 2025 19:04:26 GMT

The Periodic Table orders all Elements and is divided into groups and periods. Groups share the same amount of valence electrons Called Alkali because they combine with water to make alkaline solutions Soft and mailable, could be cut with a knife Highly reactive metals, especially towards water Harder than Alkali metals Magnesium reacts slowly with water Transition from semi-metallic () to metallic () Some elements such as gallium is almost a liquid at room temperature Transition from nonmetallic () to semi-metallic () to metallic ()
Carbon exhibits Allotropy Petroleum is a compound that contains carbon and hydrogen and is often used a source of fuel Transition from nonmetallic () semi-metallic (), to metallic () From Greek pnigein: to choke Nitric AcidNO⁻OHO Transition from nonmetallic () to semi-metallic () From calcogen: ore former Iron rusts, by being exposed to oxygen at room temperature, being turned into hydrated iron oxides From Greek halos: salt genes: former Relatively inert, non-reactive Often exists in monoatomic form

Avogadro's Number

Wed, 29 Oct 2025 18:49:05 GMT

Avogadro's number is the number of atoms in exactly 12 grams of Mole: A mole is one of Avogadro's number of any particleWhat is the mass of 6 atoms of americium (Am, 243amu)?1 mole of Am will have a mass of 243 gramsMass of one atom of Am = 243g / 243 / (60.22 * 10^23)

Octet Rule

Wed, 29 Oct 2025 18:32:46 GMT

The Octet rule states that during chemical reaction, molecules are more likely to combine to achieve full shells of electronsHydrogen, however, only needs to achieve a shell of two electronsAlso called coordinate covalent or donor-acceptor, dative bonds are ones where a single atom provides both electronsNHHBThe Formal Charge of a covalent bond can be equated byWhere is the number of valence electrons, are the lone electrons and are the bonded electrons

Ionic Bonding

Wed, 29 Oct 2025 18:28:27 GMT

Atoms are bonded when one electron is transferred from one atom to another, this creates ionsNa⁺Cl⁻

Electronegativity

Wed, 29 Oct 2025 18:27:56 GMT

In a molecule of the two electrons are shared equally However, in a mole with a difference in electronegativity the share of electrons will be differentThese bond are called "polar"

Empirical and Molecular Formulas

Wed, 29 Oct 2025 18:22:10 GMT

The molecular formula shows the number of atoms of each element in one molecule For example, the molecular formula for glucose is And its corresponding empirical formula Not all reactants will be exhausted when a reaction occurs. The reactant that is used up first is called the limiting reactant

Red-Black Tree

Tue, 28 Oct 2025 22:28:08 GMT

A red-black tree is a binary search tree that satisfies these properties Every node is either red or black The root is black Every leaf (NIL) is black If a node is red then both its children are black For each node, all simple paths form the node to decedents leaves contain the same number of black nodes A red-black tree is an extension of the binary search treeBy constraining the node colors on any simple path from the root to a lead, no path will be more than twice as long as any otherIn fact the height of a tree with nodes will at most be The height of a red-black tree with internal nodes will at most be Base case Where is the black height of a sub tree, which is the maximum height that only counts black nodes not including the rootDuring insertions, these properties can be violated 2. The root is black 4. If a node is red, then both of its children are blackBecause of this knowing the "uncle/aunt" node is more useful than knowing the parentAll insertions follow the insertion rules for binary search trees and start off red Its color changes based on any violated rules(50R is inserted) 50R is red which violates 2.(50R 50B)50B Nothing is violated(99R is inserted) 50B NIL 99R Nothing is violated(70R is inserted) 50B NIL 99R 70R (y)NIL 70R is a child of 99R which violates 4 (2b)(Right Rotate 70R 99R) 50B (y)NIL 70R 99R 99R is a child of 99R which violates 4 (3b)(70R 70B) (50B 50R) (Left Rotate 70B 50B Left) 70B 50R 99R Nothing is violated

Recursion

Tue, 28 Oct 2025 22:27:43 GMT

A method of solving a problem where the overall solution depends on solutions to smaller instances of the problemBase Case: When the answer to the problem is directly known Recursive Call: When the function calls itself to get closer to the base case Pending Operation: Remaining operations that need to be finished before the function ends Tail Recursion: When there is no remaining pending operations when the base case is reachedEach time a function call is made, a stack frame is pushed onto the stackThis repeats until the base case is reachedTheta Notation O(n) measures the worst case scenario and has a chance to break out early (n) measures a constrained scenario with no chance to break early public static int pow(int x, int y) { if (y==0) { return 1; } return x * pow(x, y - 1); } private static int powHelp(int x, int y, int result) { if (y==0) { return result; } return powHelp(x, y - 1, x * result); } public static int pow(int x, int y) { return powHelp(x, y, 1); } A String builder is an array in Java that is made out of characters. It can add or remove characters in constant time There are three different symbols to notate time complexity: Worst Case Scenario : Constant Case Scenario : Best Case ScenarioNumbers that, when arranged, can make up a triangleEx. 1, 3, 6, 10 Equation for Triangular Numbers $$ \frac{i(i+1)}{2} > n $$

Linear Structures

Tue, 28 Oct 2025 22:27:19 GMT

Stacks are linear data structures where objects are only added/removed at the topQueues are linear data structures where objects are only added in the end but only removed at the start Queues are interfaces in Java while Stacks are a classInterfaces vs Classes Interfaces define a methods and properties but can not be instantiated Classes both define state and behavior and can be instantiated Queues do not shift all elements once a object has been removed Circular queues loop around the allocated memory while simple queues do not

Time Complexity Example A

Tue, 28 Oct 2025 05:51:40 GMT

Time Complexity Example D

Tue, 28 Oct 2025 05:51:37 GMT

Map

Mon, 27 Oct 2025 21:54:20 GMT

A collection of key value pairs Every key is unique while values does not have to beget(key): put(key, value):

Tower of Hanoi Example

Mon, 27 Oct 2025 21:47:55 GMT

Derivative Rules for Vectors

Mon, 27 Oct 2025 21:26:22 GMT

Product rules for vectors are isometric to the scaler product ruleSimilarly the chain rule for vector is as follows

Optimization

Mon, 27 Oct 2025 21:23:40 GMT

To optimize on an open set, look for critical points For closed sets, use the Lagrange multiplier on boundary pointsFor optimization problems, identify The target function or desideratum The control variable () Description where appropriate Domain of admissible values Given Find the closest point to the Origin

Spherical Coordinates

Mon, 27 Oct 2025 21:23:29 GMT

Volumes can also be given using the spherical coordinates Difference between and Different system sometimes swap and , mathematically, the should match for and for The volume can be determined using spherical coordinates via the formulaSimilar to in Cylindrical Coordinates, is the constant of integration for spherical coordinates

Second Derivative Test

Mon, 27 Oct 2025 21:23:25 GMT

Given that all 2nd order partial of are continuous, in the area of a critical point . the concavity of can be determined by if and < 0, then is a local maximum if and > 0, then is a local minimum if is a saddle point Second Derivative Test Example B.excalidrawBoundaries are open if they do not contain their boundary points Closed sets do contain their boundary point Boundaries can be neither open nor closed

Lagrange Multiplier Example

Mon, 27 Oct 2025 21:23:03 GMT

Second Derivative Test Example A

Mon, 27 Oct 2025 21:21:23 GMT

Second Derivative Test Example B

Mon, 27 Oct 2025 21:21:10 GMT

Spherical Coordinates Definition

Mon, 27 Oct 2025 21:20:15 GMT

Directional Derivatives Formula

Mon, 27 Oct 2025 21:14:26 GMT

Directional Derivatives Example

Mon, 27 Oct 2025 21:14:26 GMT

Reparameterization

Mon, 27 Oct 2025 21:12:33 GMT

A new function that maps another function into the same space with different domainsA smooth reparameterization of is a scaler function such that Where and traces the same pathDerivative of inverses Given The derivative of the inverse is The arc length of a parameterized curve can be defined asMagnitude in Integration The integral of a magnitude does not equate to the magnitude of the integral

Partial Derivative

Mon, 27 Oct 2025 21:12:28 GMT

A partial derivative is deriving a function with respect to a variable, this derives only one variable while the other remains "constant" Partial Derivative of f(a, b) in respect to x $$ f_{x}(a,b) = \lim_{ h \to 0 } \frac{f(a+h,b)-f(a,b)}{h} $$ Partial Derivative of f(a, b) in respect to y $$ f_{y}(a,b) = \lim_{ h \to 0 } \frac{f(a,b+h)-f(a,b)}{h} $$Notations of Partial Derivative Partial Derivatives are can be notated in a number of ways The partial derivative can be taken multiple timesDifferentiability If all partial derivatives of a function are differentiable, then the function itself is differentiable Partial differential equations are differential equations with partial derivativesThe heat equation, for example, gives temperature as a function of position and time Mixed Partials are partial derivatives where derivatives in respect to more than one variable is taken Mixed Partial of f in respect to x and y $$ f_{xy} = \frac{\partial^2f}{\partial x \partial y} $$Order of Mixed Partials The partial is equal to Similarly, is equal to and If the relevant mixed partials exists and are continuous around the particular position then

Order of Integration

Mon, 27 Oct 2025 21:12:27 GMT

In a triple integral, bounds can be written in different ways while producing the same resultGiven , integrate over volume Since has bounds between the first integral can be set toThe second variable can be set to and can be set to

Linear Function

Mon, 27 Oct 2025 21:12:24 GMT

A linear function on has the formIn a one variable linear function , the graph is a line In a two variable linear function, the graph is a planeFor one variable functions, is differentiable at if exists, meaning that if at looks like a straight line close up, it is differentiableFor two variable functions, is differentiable at if it resembles a plane close upA scalar field is differentiable at if there is a linear function such thatWhere as The linearization of function at point is given byLinearization of functions can be used to estimate values of functionsGiven Estimate Instead of estimating values, changes can be estimated using differentials

Integrals of Vectors

Mon, 27 Oct 2025 21:12:21 GMT

Because integrals are the limits of sums, and vectors can be added together, there is a direct translation to the integral of vectors Fundamental Theorem for vectors $$ \vec{r}(t)-\vec{r}(t_{0})=\int_{t_{0}}^t\vec{r'}(u)du $$ Show that a particle that changes direction has nonzero acceleration Suppose The velocity can be defined as And since The velocity can not change direction if A particle initially at rest at the origin is subject to an acceleration For Since Then is equal to For

Gradient Decent

Mon, 27 Oct 2025 21:12:17 GMT

The minimization (decent) or maximization (accent) of a cost/loss function of several variables

Curvature

Mon, 27 Oct 2025 21:12:08 GMT

The curvature of a function is how much the direction of a tangent line changes throughout the functionA curve is called a smooth curve when Smooth curves contains a well-defined unit tangent vectorCurvature can be defined using as followsThe curvature can also be given using

Changing Coordinates

Mon, 27 Oct 2025 21:12:03 GMT

Areas can also be integrated via Consider a polar rectangleArea of the rectangle can be determined the following equation Given where , can be expressed asThe area using Can be given as

Optimization

Mon, 27 Oct 2025 20:21:54 GMT