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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Functions of Several Variables">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Sun Mar 30 13:32:28 1997}
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\begin{document}


\section{Functions of Several Variables\label{Functions of Several Variables}
}

\begin{definition}
Let $D$ be a subset of the $xy$-plane, written $D\subset R^{2}$. A \textbf{%
function }$f$\textbf{\ of two variables} is a rule that assigns to each
ordered pair $\left( x,y\right) $ in $D$ a unique real number denoted by $%
f(x,y)$. The set $D$ is the domain of $f$ and its range is the set of values
that $f$ assumes. That is , 
\[
\left\{ f(x,y)\ \left| \ \left( x,y\right) \in D\right. \right\} 
\]
\end{definition}

\begin{definition}
If $f$ is a function of two variables with domain $D$, the\textbf{\ graph }%
of $f$ is the set 
\[
S=\left\{ (x,y,z)\in R^{3}\left| \text{ }z=f(x,y),(x,y)\in D\right. \right\} 
\]
Just as the graph of a function $f$ of one variable is a curve $C$ with
equation $y=f(x)$, so the graph of a function $f$ of two variables is a
surface $S$ with equation $Z=f(x,y)$. You can visualize the graph $S$ of $f$
as lying directly above or below its domain $D$ in the $xy$-plane.
\end{definition}

\subsubsection{Level Curves\label{Level Curves}}

\begin{definition}
The \textbf{level curves }of a function $f$ of two variables are the curves
with equations $f(x,y)=k$, where $k$ is a constant (in the range of $f\,$).
\end{definition}

A level curve $f(x,y)=k$ is the locus of all points at which $f$ takes on a
given value $k$. In other words, it shows where the graph of $f$ has height
(or altitude) $k$.

The level curves $f(x,y)=k$ are just the traces of the graph of $f$ in the
horizontal plane $z=k$ projected down to the $xy$-plane. So if you draw the
level curves of a \ function and visualize them being lifted up to the
surface at the indicated height, then you\ can mentally piece together a
picture of the graph. The surface is steep where the level curves are close
together. It is less steep where they are farther apart.

\subsubsection{Functions of Three or More Variables\label%
{Functions of Three or More Variables}}

A \textbf{function of three variables}, $f$, is a rule that assigns to each
ordered triple$\left( x,y,z\right) $ in a domain $D\subset R^{3}$ a unique
real number denoted by $f(x,y,z)$. For instance, the temperature $T$ depends
on the time $t$ as well as on the longitude $x$ and longitude $y$, so we
could write $T=f(x,y,t)$.

It is possible to visualize a function $f$ of three variables by an arrow
diagram but not by its graph, since that would lie in a four-dimensional
space. However you can gain some insight into $f$ by examining its \textbf{%
level surfaces}, which are the surfaces with equations $f(x,y,z)=k,$ where $%
k $ is a constant. If the point $\left( x,y,z\right) $ moves along a level
surface, the value of $f(x,y,z)$ remains fixed.

Functions of any number of variables can also be considered. A \textbf{%
function of n variables} is a rule that assigns a number $%
z=f(x_{1},x_{2},\ldots ,x_{n})$ to an n-tuple $(x_{1},x_{2},\ldots ,x_{n})$
of real numbers.

The notation 
\[
f:\text{ }D\subset R^{n}\rightarrow R 
\]
is used to signify that $f$ is a real-valued function whose domain $D$ is a
subset of $R^{n}$.

Sometimes you will use vector notation to write functions more compactly: If 
$x=\langle x_{1},x_{2},\ldots ,x_{n}\rangle $, you can write $f(x)$ in place
of $f(x_{1},x_{2},\ldots ,x_{n})$.

In view of the one-to-one correspondence between points $(x_{1},x_{2},\ldots
,x_{n})$ in $R^{n}$ and their position vectors $x=\langle x_{1},x_{2},\ldots
,x_{n}\rangle $ in $V_{n}$, we have three ways of looking at a function $%
f:R^{n}\rightarrow R$:

\begin{enumerate}
\item  As a function of n real\ variables $x_{1},x_{2},\ldots ,x_{n}$

\item  As a function of a single point variable $\left( x_{1},x_{2},\ldots
,x_{n}\right) $

\item  As a function of a single vector variable $x=\langle
x_{1},x_{2},\ldots ,x_{n}\rangle $
\end{enumerate}

\subsection{Related Scientific Notebook Topics}

{\small \hyperref{Defining functions}{}{}{../../Help/DM3-2.tex#Define,
function}}

{\small \hyperref{Plot 3D, cylindrical}{}{}{../../Help/DM6-6.tex#Cylindrical
coordinates}}

{\small \hyperref{Plot 3D, spherical}{}{}{../../Help/DM6-6.tex#Spherical
coordinates}}

{\small \hyperref{Plot 3D, tube}{}{}{../../Help/DM6-6.tex#Curves in space}}

\noindent \hrulefill

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\noindent \hrulefill

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