So far, we have examined only functions of two variables. However, many real-world situations require functions of three or more variables. Consider these examples:
[latex]f(x,y,z)=x^{2}-2xy+y^{2}+3yz-z^{2}+4x-2y+3z-6[/latex] (a polynomial in three variables)
In the first function, [latex](x,\ y,\ z)[/latex] represents a point in space, and the function [latex]f[/latex] maps each point to a fourth quantity, such as temperature or wind speed. In the second function, [latex](x,\ y)[/latex] represents a point in the plane and [latex]t[/latex] represents time. This function might map a point in the plane to a third quantity (such as pressure) at a given time [latex]t[/latex].
The method for finding the domain of a function of more than two variables follows the same principles as for functions of one or two variables.
Find the domain of each of the following functions:
For the function [latex]f\,(x,\ y,\ z)=\frac{3x-4y+2z}{\sqrt{9-x^{2}-y^{2}-z^{2}}}[/latex] to be defined (and be a real value), two conditions must hold:
The denominator cannot be zero
The radicand cannot be zero
Combining these conditions leads to the inequality
[latex]9-x^{2}-y^{2}-z^{2}\,>\,0[/latex].
Moving the variables to the other side and reversing the inequality gives the domain as
which describes a ball of radius [latex]3[/latex] centered at the origin. (Note: The surface of the ball is not included in this domain.)
For the function [latex]g\,(x,\ y,\ t)=\frac{\sqrt{2t-4}}{x^{2}-y^{2}}[/latex] to be defined (and be a real value), two conditions must hold:
The radicand cannot be negative.
The denominator cannot be zero.
Since the radicand cannot be negative, this implies [latex]2t-4\,\geq\,0[/latex], and therefore that [latex]t\,\geq\,2[/latex]. Since the denominator cannot be zero, [latex]x^{2}-y^{2}\neq{0}[/latex], or [latex]x^{2}\neq{y}^{2}[/latex], which can be rewritten as [latex]y\neq\pm{x}[/latex], which are the equations of two lines passing through the origin. Therefore, the domain of [latex]g[/latex] is
Remember the key restrictions when finding domains:
Square roots require nonnegative radicands
Denominators cannot equal zero
Logarithms require positive arguments
Level Surfaces
Functions of two variables have level curves, which appear as curves in the [latex]xy[/latex]-plane. When the function has three variables, these curves extend into three dimensions and become level surfaces.
level surfaces
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].
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].
The level surface is defined by the equation [latex]4x^{2}+9y^{2}-z^{2}=1[/latex]. This equation describes a hyperboloid of one sheet as shown in the following figure.
Figure 1. A hyperboloid of one sheet with some of its level surfaces.
Level surfaces are the three-dimensional analog of level curves. Just as level curves show where a two-variable function has constant value, level surfaces show where a three-variable function has constant value. Different values of [latex]c[/latex] produce different level surfaces.
