# Lecture 009

## Function with Two Variables

Function with Two Variables: $z(x, y) = ...$

• $(x, y) = (x', y') \implies z(x, y) = z(x', y')$

• Domain: what $(x, y)$ can be

• Range: what $z$ can be

Level Curve: level curve at $z=c$ is the curve in $R^2$ of $z = f(x, y)$.

Contor Map: several labeled level curves on same coordinate axes (ie. multiple level curves in the same direction)

Vertical Trace: making one independent (instead of dependent) variable constant

## Functions with Three Variables

Functions with Three Variables: $w = f(x, y, z)$

• a hypersurface in $\mathbb{R}^4$

• visualize by Level Surfaces

If $z = f(x, y)$, the limit of $f(x, y)$ as $(x, y)$ approaches $(a, b)$ is $C$, denoted $\lim_{(x, y) \rightarrow (a, b)} f(x, y) = C$ provided that for all $\epsilon$ so that when $0 < \text{dist}((x, y), (a, b)) < S$, $|f(x, y) - C| < S$

## Functions with More Variables

$w = f(x, y, z) \in \mathbb{R}^4$

• visualize with level surface in $\mathbb{R}^3$

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