A function is a * function of several variables* if --- that is, if there is more than one
input variable.

For example a function is
a * parametrized surface* in . Here's a picture of

Or consider a function . Its
* graph* is a surface in . Here's a picture of the graph of :

Functions of several variables occur in many real world situations --- in fact, most measurable quantities depend on many factors or variables. For example, the temperature at a point in space may be a function of its coordinates . The ideal gas law relates the pressure p, the volume V, and the temperature T of an ideal gas. I can regard any one of these three variables as a function of the other two; for example, writing views p as a function of V and T.

* Example.* A function is defined by

Evaluate , , and .

You substitute values into a function of several variables in the obvious way. For instance, to evaluate , I set , , and in the formula for f:

Likewise,

* Definition.* For a function :

(a) The * domain* is the set of all points in
where f is defined.

(b) The * image* (or the *
range*) is the set of all outputs of f in .

* Remark.* I'm following the usual convention:
For functions to
refer to the set of points in where the function is defined as the * domain* of the function. Thus, for the function
, the
"domain" is the set of points such that , i.e. .

In more advanced courses, a more precise definition of a * function* requires that the "domain" be
included as part of the function's definition. In that context, what
we're calling the "domain" is referred to as the * natural domain* (the "biggest possible
set" where the function is defined).

* Example.* A function of 2 variables is defined
by

Describe the set of points for which f is undefined. What is the * domain* of f?

is undefined when the denominator is 0:

Thus, f is undefined for points on either of the lines or .

The domain is the set of points which are *not* on or -- i.e. the points such that .

* Example.* A function of 2 variables is defined
by

Describe the set of points for which f is undefined. What is the * domain* of f?

The natural log function is undefined for inputs which are less than or equal to 0. So will be undefined if

That is, f is undefined at points where . They are the points lying on or above the line :

The domain is the set of points *below* the line .

* Example.* A function of 2 variables is defined
by

Describe the set of points for which f is undefined. What is the * domain* of f?

Since the square root of a negative number is undefined, is undefined for

That is, f is undefined at points lying outside the unit circle .

The domain is the set of points on or inside the unit circle .

* Example.* Describe the image (or the range) of
the function .

The image of a function is the set of outputs. What are the outputs of the inverse tangent function ?

For any x and y,

I know that attains every value between and , and if I set I have .

Therefore, the image of f is .

A function of several variables can be pictured in many ways. For
example, you can draw the * graph* of a function
of 2 variables . Because
plotting points in 3 dimensions is tedious and difficult, you'd
probably use software to draw the graph.

for functions , you
may also get a "picture" of the function by drawing its * level sets*. A level set for f is obtained by
setting , where c is a constant. By using
different values for c, you get a picture of the "levels"
of the function.

You may have seen * topographic maps*, where the
level curves are referred to as * contour lines*.
They represent lines along which the altitude ("z") is
constant:

* Example.* Sketch the graph of , and some of the contour lines.

Here's the graph of the function, produced by a computer:

I can use a computer to sketch the level curves, but this example is simple enough that I'll analyze it first. I get level curves by setting z to specific numbers and graphing the curves I get.

You can see the level curves are a family of parabolas.

For a function of 3 variables , setting for various numbers c produces * level surfaces*. If you interpret as the temperature at a point in space, then a level surface is the set of points in space where the
temperature is c.

* Example.* Describe some level surfaces for the
function .

The level surfaces are paraboloids opening along the x-axis.

Copyright 2018 by Bruce Ikenaga