Scalar Field

A scalar field is a function of several variables of the type

𝑓 : ℝ 𝑛 \to ℝ

Which takes in a multi-dimensional vector and outputs a scalar
With the domain

𝑑 𝑜 𝑚 𝑓 = {(𝑥, 𝑦) \in ℝ 2 : 𝑓 = (𝑥, 𝑦) \in ℝ}

And the image

𝑖 𝑚 𝑓 = {(𝑥, 𝑦) \in ℝ : 𝑓 = (𝑥, 𝑦) \in 𝑑 𝑜 𝑚 𝑓}

The level set for a scalar field $𝑓$ is the set of all input values that output $𝑘$

𝑓 - 1 (𝑘) = 𝑥 \in 𝑑 𝑜 𝑚 𝑓 : 𝑓 (𝑥) = 𝑘)

The level set of $𝑓 (𝑥, 𝑦) = 𝑥 2 + 𝑦 2$ is a circle

$𝑓 (𝑥, 𝑦) = 𝑥 2 + 2 𝑥 𝑦 + 𝑦 2 9$

Check continuity (plug it in)
Check a few paths
1. Any two paths differ then the limit DNE
2. The limit can not be proved to exist via exhaustion
Try Polar Coordinates

Solving limits via polar cordinate

l i m (𝑥, 𝑦) \to 0 𝑥 2 𝑦 𝑥 2 + 𝑦 2

l i m 𝑟 \to 0 + (𝑟 c o s (𝜃)) 2 𝑟 s i n (𝜃) 𝑟 2

l i m 𝑟 \to 0 + 𝑟 c o s (𝜃) 2 s i n (𝜃)