# Section12.1Introduction to Multivariable Functions¶ permalink

##### Definition12.1.1Function of Two Variables

Let $D$ be a subset of $\mathbb{R}^2\text{.}$ A function $f$ of two variables is a rule that assigns each pair $(x,y)$ in $D$ a value $z=f(x,y)$ in $\mathbb{R}\text{.}$ $D$ is the domain of $f\text{;}$ the set of all outputs of $f$ is the range.

##### Example12.1.2Understanding a function of two variables

Let $z=f(x,y) = x^2-y\text{.}$ Evaluate $f(1,2)\text{,}$ $f(2,1)\text{,}$ and $f(-2,4)\text{;}$ find the domain and range of $f\text{.}$

Solution
##### Example12.1.3Understanding a function of two variables

Let $\ds f(x,y) = \sqrt{1-\frac{x^2}9-\frac{y^2}4}\text{.}$ Find the domain and range of $f\text{.}$

Solution

# Subsection12.1.1Graphing Functions of Two Variables

The graph of a function $f$ of two variables is the set of all points $\big(x,y,f(x,y)\big)$ where $(x,y)$ is in the domain of $f\text{.}$ This creates a surface in space.

One can begin sketching a graph by plotting points, but this has limitations. Consider Figure 12.1.5(a) where 25 points have been plotted of $\ds f(x,y) = \frac1{x^2+y^2+1}\text{.}$ More points have been plotted than one would reasonably want to do by hand, yet it is not clear at all what the graph of the function looks like. Technology allows us to plot lots of points, connect adjacent points with lines and add shading to create a graph like Figure 12.1.5b which does a far better job of illustrating the behavior of $f\text{.}$

While technology is readily available to help us graph functions of two variables, there is still a paper–and–pencil approach that is useful to understand and master as it, combined with high–quality graphics, gives one great insight into the behavior of a function. This technique is known as sketching level curves.

# Subsection12.1.2Level Curves

It may be surprising to find that the problem of representing a three dimensional surface on paper is familiar to most people (they just don't realize it). Topographical maps, like the one shown in Figure 12.1.8, represent the surface of Earth by indicating points with the same elevation with contour lines. The elevations marked are equally spaced; in this example, each thin line indicates an elevation change in 50ft increments and each thick line indicates a change of 200ft. When lines are drawn close together, elevation changes rapidly (as one does not have to travel far to rise 50ft). When lines are far apart, such as near “Aspen Campground,” elevation changes more gradually as one has to walk farther to rise 50ft.

Given a function $z=f(x,y)\text{,}$ we can draw a “topographical map” of $f$ by drawing level curves (or, contour lines). A level curve at $z=c$ is a curve in the $x$-$y$ plane such that for all points $(x,y)$ on the curve, $f(x,y) = c\text{.}$

When drawing level curves, it is important that the $c$ values are spaced equally apart as that gives the best insight to how quickly the “elevation” is changing. Examples will help one understand this concept.

##### Example12.1.9Drawing Level Curves

Let $\ds f(x,y) = \sqrt{1-\frac{x^2}9-\frac{y^2}4}\text{.}$ Find the level curves of $f$ for $c=0\text{,}$ $0.2\text{,}$ $0.4\text{,}$ $0.6\text{,}$ $0.8$ and $1\text{.}$

Solution
##### Example12.1.13Analyzing Level Curves

Let $\ds f(x,y) = \frac{x+y}{x^2+y^2+1}\text{.}$ Find the level curves for $z=c\text{.}$

Solution

# Subsection12.1.3Functions of Three Variables

We extend our study of multivariable functions to functions of three variables. (One can make a function of as many variables as one likes; we limit our study to three variables.)

##### Definition12.1.17Function of Three Variables

Let $D$ be a subset of $\mathbb{R}^3\text{.}$ A function $f$ of three variables is a rule that assigns each triple $(x,y,z)$ in $D$ a value $w=f(x,y,z)$ in $\mathbb{R}\text{.}$ $D$ is the domain of $f\text{;}$ the set of all outputs of $f$ is the range.

Note how this definition closely resembles that of Definition 12.1.1.

##### Example12.1.18Understanding a function of three variables

Let $\ds f(x,y,z) = \frac{x^2+z+3\sin(y) }{x+2y-z}\text{.}$ Evaluate $f$ at the point $(3,0,2)$ and find the domain and range of $f\text{.}$

Solution

# Subsection12.1.4Level Surfaces

It is very difficult to produce a meaningful graph of a function of three variables. A function of one variable is a curve drawn in 2 dimensions; a function of two variables is a surface drawn in 3 dimensions; a function of three variables is a hypersurface drawn in 4 dimensions.

There are a few techniques one can employ to try to “picture” a graph of three variables. One is an analogue of level curves: level surfaces. Given $w=f(x,y,z)\text{,}$ the level surface at $w=c$ is the surface in space formed by all points $(x,y,z)$ where $f(x,y,z)=c\text{.}$

##### Example12.1.19Finding level surfaces

If a point source $S$ is radiating energy, the intensity $I$ at a given point $P$ in space is inversely proportional to the square of the distance between $S$ and $P\text{.}$ That is, when $S=(0,0,0)\text{,}$ $\ds I(x,y,z) = \frac{k}{x^2+y^2+z^2}$ for some constant $k\text{.}$

Let $k=1\text{;}$ find the level surfaces of $I\text{.}$

Solution

In the next section we apply the concepts of limits to functions of two or more variables.

# Subsection12.1.5Exercises

Terms and Concepts

In the following exercises, give the domain and range of the multivariable function.