Functions of More Than Two Variables

Learning Outcomes

Recognize a function of three or more variables and identify its level surfaces.

So far, we have examined only functions of two variables. However, it is useful to take a brief look at functions of more than two variables. Two such examples are

$f\,(x,\ y,\ z)=x^{2}-2xy+y^{2}+3yz-z^{2}+4x-2y+3x-6$ (a polynomial in three variables)

and

$g\,(x,\ y,\ t)=(x^{2}-4xy+y^{2})\sin{t}-(3x+5y)\cos{t}$ .

In the first function, $(x,\ y,\ z)$ represents a point in space, and the function $f$ maps each point in space to a fourth quantity, such as temperature or wind speed. In the second function, $(x,\ y)$ can represent a point in the plane, and $t$ can represent time. The function might map a point in the plane to a third quantity (for example, pressure) at a given time $t$ . The method for finding the domain of a function of more than two variables is analogous to the method for functions of one or two variables.

Example: Domains for Functions of Three Variables

Find the domain of each of the following functions:

$f\,(x,\ y,\ z)=\frac{3x-4y+2z}{\sqrt{9-x^{2}-y^{2}-z^{2}}}$
$g\,(x,\ y,\ t)=\frac{\sqrt{2t-4}}{x^{2}-y^{2}}$

Show Solution

For the function $f (x, y, z) = \frac{3 x - 4 y + 2 z}{\sqrt{9 - x^{2} - y^{2} - z^{2}}}$ to be defined (and be a real value), two conditions must hold:
1. The denominator cannot be zero
2. The radicand cannot be zero
Combining these conditions leads to the inequality

$9-x^{2}-y^{2}-z^{2}\,>\,0$ .

Moving the variables to the other side and reversing the inequality gives the domain as

$\text{domain}\ (f)=\{(x,\ y,\ z)\in\mathbb{R}^{3}|x^{2}+y^{2}+z^{2}\,<\,9\}$ ,

which describes a ball of radius $3$ centered at the origin. (Note: The surface of the ball is not included in this domain.)
For the function $g (x, y, t) = \frac{\sqrt{2 t - 4}}{x^{2} - y^{2}}$ to be defined (and be a real value), two conditions must hold:
1. The radicand cannot be negative.
2. The denominator cannot be zero.
Since the radicand cannot be negative, this implies $2t-4\,\geq\,0$ , and therefore that $t\,\geq\,2$ . Since the denominator cannot be zero, $x^{2}-y^{2}\neq{0}$ , or $x^{2}\neq{y}^{2}$ , which can be rewritten as $y\neq\pm{x}$ , which are the equations of two lines passing through the origin. Therefore, the domain of $g$ is

$\text{domain}\ (g)=\{(x,\ y,\ t)|y\neq\pm{x},\,t\,\geq\,2\}$ .

Try It

Find the domain of the function $h\,(x,\ y,\ t)=(3t-6)\,\sqrt{y-4x^{2}+4}$ .

Show Solution

$\text{domain}\,(h)=\{(x,\ y,\ t)\in\mathbb{R}^{3}|y\,\geq\,4x^{2}-4\}$

Functions of two variables have level curves, which are shown as curves in the $xy$ -plane. However, when the function has three variables, the curves become surfaces, so we can define level surfaces for functions of three variables.

Definition

Given a function $f\,(x,\ y,\, z)$ and a number $c$ in the range of $f$ a is defined to be the set of points satisfying the equation $f\,(x,\ y,\ z)=c$ .

Example: Finding a Level Surface

Find the level surface for the function $f\,(x,\ y,\ z)=4x^{2}+9y^{2}-z^{2}$ corresponding to $c=1$ .

Show Solution

The level surface is defined by the equation $4x^{2}+9y^{2}-z^{2}=1$ . This equation describes a hyperboloid of one sheet as shown in the following figure.

This figure consists of four figures. The first is marked c = 0 and consists of a double cone (that is, two nappes) with their apex at the origin. The second is marked c = 1 and it looks remarkably similar to the first except that there is no apex at which the cones meet: instead, the two nappes are connected. Similarly, the next figure marked c = 2 has the two nappes connect, but this time their connection is larger (that is, the radius of their connection is greater). The final figure marked c = 3 also has the two nappes connect in an even larger fashion.

Figure 1. A hyperboloid of one sheet with some of its level surfaces.

Try It

Find the equation of the level surface of the function

$g\,(x,\ y,\ t)=x^{2}+y^{2}+z^{2}-2x+4y-6z$

corresponding to $c=2$ , and describe the surface, if possible.

Show Solution

$(x-1)^{2}+(y+2)^{2}+(z-3)^{2}=16$ describes a sphere of radius $4$ centered at the point $(1,-2, 3)$ .

Watch the following video to see the worked solution to the above Try It

You can view the transcript for “CP 4.5” here (opens in new window).

Module 4: Differentiation of Functions of Several Variables