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

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

and

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

In the first function, [latex](x,\ y,\ z)[/latex] represents a point in space, and the function [latex]f[/latex] maps each point in space to a fourth quantity, such as temperature or wind speed. In the second function, [latex](x,\ y)[/latex] can represent a point in the plane, and [latex]t[/latex] can represent time. The function might map a point in the plane to a third quantity (for example, pressure) at a given time [latex]t[/latex]. 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:

  1. [latex]f\,(x,\ y,\ z)=\frac{3x-4y+2z}{\sqrt{9-x^{2}-y^{2}-z^{2}}}[/latex]
  2. [latex]g\,(x,\ y,\ t)=\frac{\sqrt{2t-4}}{x^{2}-y^{2}}[/latex]

Try It

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

Functions of two variables have level curves, which are shown as curves in the [latex]xy[/latex]-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 [latex]f\,(x,\ y,\, z)[/latex] and a number [latex]c[/latex] in the range of [latex]f[/latex] a level surface of a function of three variables is defined to be the set of points satisfying the equation [latex]f\,(x,\ y,\ z)=c[/latex].

Example: Finding a Level Surface

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

Try It

Find the equation of the level surface of the function

[latex]g\,(x,\ y,\ t)=x^{2}+y^{2}+z^{2}-2x+4y-6z[/latex]

corresponding to [latex]c=2[/latex], and describe the surface, if possible.

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