Graphing functions of several variables

In studying single-variable functions, we have seen that plotting a graph often displays the important features of a function from which we can obtain useful information. Graphs are also useful for visualising functions of two variables. Consider a single-variable function, $y=g(x)$ and a two-variable function, $z=$ $f(x, y)$. In the same way that an equation $y=g(x)$ leads to plotting ordered pairs $(x, y)$ in the Cartesian plane (see Fig. [1.1), the equation, $z=f(x, y)$ leads to plotting ordered triples $(x, y, z)$ in a three-dimensional space (see Fig. 1.2). We can extend this to functions of three variables, e.g. $w=h(x, y, z)$ to represent four-dimensional surfaces.

Figure 1.1: Graph of $y=g(x)$ for $g(x)=x^{3}$.

Plotting functions of two variables by hand is often a tedious and difficult task; fortunately we rely on computer software (e.g. MATLAB) which allows us to explore functions graphically from different perspectives. Of course it is not possible to draw the graph of a function of more than two variables.

Before we move on, let us briefly discuss the domain, $D$ of functions of more than one variable. Recall for a single-variable function, say $y=g(x)$, the domain consists of all the values of $x$ that we can plug into the function and get back a real number. This implies that the domain of a function of a single variable is an interval (or intervals) of values from the number line or one-dimensional space. For a two-variable function, say $z=f(x, y)$, the domain is given by a two-dimensional space consisting of all the coordinate

Figure 1.2: Graph of $z=f(x, y)$ for $f(x, y)=\cos \sqrt{x^{2}+y^{2}}$.

pairs, $(x, y)$ that we can plug into the function and get back a real number. If we are given a function $f(x, y)$ and the domain is not specified, then it is understood that the domain of $f$ is the set of all pairs $(x, y)$ for which $f$ is a well-defined number.

Example 1.1 Find the domain of the following function and evaluate $f(3,2)$ :

Solution The expression for $f$ gives a real number if the denominator is not 0 and if the quantity under the square root is nonnegative 1

The denominator is not equal to 0 if $x \neq 1$ and hence, the points on the line $x=1$ must be excluded from the domain. For the numerator, since $x+y+1 \geq 0$, then the inequality $y \geq-x-1$ describes the points that lie on or above the line $y=-x-1$. The domain of $f$ is given by:

Finally, to evaluate $f(3,2)$, we plug in $x=3$ and $y=2$ in the expression for $f$ :

${ }^{1}$ note that the term nonnegative is used for a quantity which is either zero or positive, i.e, $\geq 0$. Next, in order to analyse graphs, we define and discuss traces and contour maps.

Traces

Traces are the curves which are obtained by intersecting the graph with planes parallel to a coordinate plane. We define three types of traces:

1. Horizontal trace at height $\boldsymbol{k}$ : the intersection of the graph with the horizontal plane $z=k$, consisting of points $(x, y, k)$ such that $f(x, y)=k$ (see Fig. 1.31)
2. Vertical trace in the plane $x=a$ : the intersection of the graph with the vertical plane at $x=a$ [see Fig. 1.4(a)
3. Vertical trace in the plane $\boldsymbol{y}=\boldsymbol{b}$ : the intersection of the graph with the vertical plane at $y=b$ [see Fig. 1.4(b)

Figure 1.3: Graph of $z=f(x, y)$ intersecting with the horizontal plane at $k=0.8$.

Contour maps

The level curve of a function, $f(x, y)$ is defined as the curve in the $x y$-plane with equation $f(x, y)=k$, i.e. it shows where the graph of $f$ has height $k$. A collection of such curves is called the contour diagram or contour map of the function $f(x, y)$. It is useful to look at the relationship between level curves and horizontal traces. The level curves $f(x, y)=k$ are the traces of the graph $f$ in the horizontal plane, $z=k$ projected down to the $x y$-plane. Figure 1.5 shows a number of horizontal traces and their projections on the $x y$-plane; as an example a horizontal plane is highlighted at $z=22$ while the dotted,

Figure 1.4: Vertical traces parallel to (a) $y z$-plane with trace shown with a thick red line at $x=-15$ (b) $x z$-plane with trace shown with a thick red line at $y=-17$.

vertical lines show the ptojection of that trace to the $k=22$ level curve on the $x y$-plane.

Figure 1.5: Horizontal traces and level curves $f(x, y)=k$ at various values of $k$. A horizontal plane and trace are highlighted at $z=22$ with the projected level curve shown at $k=22$ on the $x y$-plane. Example 1.2 Sketch the level curves for $k=0,1,2$ of the function

Solution The level curves are given by $f(x, y)=k$ or, equivalently,

These curves represent concentric circles with centre at $(0,0)$ and radius $r=\sqrt{9-k^{2}}$. Figure 1.6 shows the level curves at $k=0,1,2$ (labelled).

Figure 1.6: Contour map of $f(x, y)=\sqrt{9-x^{2}-y^{2}}$ showing level curves for $k=0,1,2$.