Multivariable Functions

Multivariable Functions

Observation: We can extend the single-valued concept of functions. A function $f$ assigns a value $\vec{x}\in {\mathbb{R}}^m$ to a value $\vec{y}\in {\mathbb{R}}^n$. $f$ is single-valued if $m=1$ and vector-valued if $m>1$.
Example: $f:(x,y,z)\mapsto (x^2+y^2+z^2{)}^{-3/2}$

Definition: $f:U\subset {\mathbb{R}}^n\to \mathbb{R}$ means that $f$ is a real-valued function of n variables with domain U

Definition: The graph of $f:U\subset {\mathbb{R}}^n\to {\mathbb{R}}^m$ is the subset of ${\mathbb{R}}^{n+m}$ consisting of all the points $(x_1,\dots ,x_n,f(x_1,\dots x_n)\in {\mathbb{R}}^m|(x_1,\dots ,x_n)\in U)$

Visualizing Functions

Observation: These graphs are easy enough to visualize when $f:{\mathbb{R}}^2\to \mathbb{R}$ or $f:\mathbb{R}\to \mathbb{R}$ but visualizing higher dimensions is problematic.

Definition: Let $f:U\subset {\mathbb{R}}^n\to \mathbb{R}$ and let $c\in \mathbb{R}$, then the level set of value $c$ is defined to be the set of those points $\vec{x}\in U$ at which $f(\vec{x})=c$. If $n=2$ we call this a level curve and if $n=3$ we call this a level surface.

Example: Say $f(x,y,z)=x^2+y^2+z^2$, the set where $x^2+y^2+z^2=c$ is a level set for $f$.

This case illustrates something interesting, that graphs can often be conceptualized just fine projected onto a lower dimension (think a contour map).

Definition: A section of a graph of $f$ is the intersection of the graph with a vertical plane. This information can complement the level set in visualizing functions.