Let us begin with an example.

Consider the function defined for every in . Domain of this function is the set

(1)

and the range of is all numbers in that is

(2)

This function can be shown as

We also know the familiar graph of this function

Now what if we are given a function which depends on two variables, say and Consider the function defined on the unit disk. The domain of this function is the set

(3)

The range of is the set of all possible outputs of that is

(4)

This function can be shown as

The graph of is shown below. Note that the origin is the lowest point on the graph of

The function in example 2 is said to be a single-valued function of two variables, single output and a pair of inputs. Let us look at another example.

Define defined on the first quadrant of the unit disk. The domain of this function is the set

(5)

The range of is

(6)

This function can be shown as

And the graph in -space is

One can extend what we have done in previous examples to define functions of more than two variables. Such functions are called functions of several variables. In this case, it will not be possible to graph them but we still can get a sense of how they behave. To begin, consider the following example.

Consider the function in example 2. If we fix , then is a horizontal cut in the graph of at We like to see the shadow of such cut in -plane. In fact, for any fixed in we get a cut of the graph, say or The shadows of these cuts are shown below. If one cuts the graph of the function at of course nothing is obtained since is not in the range of

Such shadow curves are called level curves or in general contours.

Consider the function We are interested in seeing the contours for for . Start by graphing each of

(7)

We draw the contours associated to when

The contours of a function of three variables is called level surfaces. This is due to the fact that for and for some the contour is a surface in -space.

We are given defined for any point in -space. Each level surface of is a sphere, centered at the origin. For example, given we obtain a level surface of which is that is a sphere of radius