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We introduce functions that take vectors or points as inputs and output a
The world is constantly changing. Sometimes this change is very slow, other times
it is shockingly fast. Consider Meteor Crater in northern Arizona.
This area was once grasslands and woodlands inhabited by bison, camels, wooly
mammoths, and giant ground sloths. During the Pleistocene epoch, a meteor only
meters in diameter collided with the Earth and this changed very quickly. The
collision released around joules of energy, comparable to the energy released by a
large nuclear weapon. A fireball extended out kilometers from the center of the
impact, destroying all life in its wake. It is estimated it took one hundred years for
the local plant and animal life to repopulate the area. Fifty thousand years
later, the remains of the impact crater are still intact on our ever-changing
To help us understand events like these, we need to precisely describe what we are
observing (in this case, the crater). To do this we use a contour map, often called a
In essence, we are looking at the crater from directly above, and each curve in the
map above represents a fixed, constant height. Mathematically, a contour map
illustrates a function of two variables. We will now define a more general
case of a function of variables. These are often called functions of several
Let be a subset of . A function of variables, also called a function of several
variables, with domain is a relation that assigns to every ordered -tuple in a
unique real number in . We denote this by each of the following types of
The range of is the set of all outputs of . It is a subset of , not .
Find the domain and range of .
Here, the domain is All points in with and
and the range is .
The relationship from the previous example can be described more succinctly by the
equation which is the notation that we will use most frequently when describing
In this text, we will use an upper-case letter to denote a function of several
Often, we will not specify the domain of a function in order to shorten its description.
Unless otherwise specified, we will take the domain of a given function on to be the
set of all ordered tuples in for which the given formula is defined . We are familiar
with this concept from one-variable calculus, where we would see a function defined
by a formula such as and take its domain to be . In our example , we take its domain
to be .
Let’s investigate a few functions of two variables, .
Consider What is ?
What is the domain of ? Since we have not specified the domain, we take it to be
the set of all vectors allowable as inputsoutputs for . Because of the logarithm, we need such that
The observant reader may note that this inequality describes the interior of a circle of
radius centered at in the -plane since we can write
While the domain may not always be easy to visualize, it is and excellent practice
and often insightful to try such visualization.
What is the range of ? The range is the set of all possible inputoutput values. If we visualize the graph of , we can see that the logarithm function outputs
all values in . However, the input for our logarithm function is not any value of , but
any value of . Since the and terms are squared and then subtracted from , the
largest possible value of occurs where and , in which case . Notice that we must also
have in order to calculate the logarithm.
What do these calculations mean for the range of ? In general, the logarithm is an increasingdecreasing function of its input, meaning that as the input gets larger, the output gets largersmaller. In other words, the largest value of gives us the largest possible value of . We
similarly find smaller values of by plugging in smaller values of . We have determined
that the values of which make sense for this problem are those in the interval , and
so evaluating the logarithm on this interval gives us that the range is the interval .
Consider this geometric example.
The volume of a cylinder with base radius and
height is given by We can now think of the volume of a cylinder as a function of two
variables, and Find the domain and the range of .
By requiring that the radius and
height be nonnegative, we find that the domain is Points in where and , or
in set notation . The range is: . The domain represents the set of all possible nonnegative radii and heights of the
cylinder, and the range represents the set of all possible volumes that a cylinder
Visualizing functions of several variables
There are many ways to interpret a function of several variables. Two very common
ways to do this are to consider the surface obtained by graphing the function or to
look at what we will call the level sets of our function.
Recall that given a function of a single variable, we can consider the equation ,
which allows us to visualize the function as the set of all points in the -plane. To do
this, we pick an -value in the domain, and then the corresponding -coordinate is
given by .
Given a function of two variables, we can take the same approach. We’ll
consider the set of all points in -space where . By choosing a point in the
domain of the function, the corresponding -coordinate will be given by .
Thus, one way of visualizing the function is to consider the equation and consider
the set of all of the points in the -space that satisfy this criteria. We can then
interpret that the function assigns a height to each point in its domain. Be very
careful with this way of visualizing the function, however! The “height” can
sometimes be negative, while we tend to almost always visualize a positive
We do not always interpret the output as a height. For instance, we might want to
talk about a density function for a region in , and define a function by the density at
each point in the region. We can still graph , but the -values now should be
interpreted as densities. Be careful to keep the meaning of the function in
To make a sketch of a surface, we can specify many locations in the -plane
(by picking many different values for and ), and plot the corresponding
-values. While this is tedious to do by hand, computers can do it very easily.
For example, if we consider the function , we can evaluate the function at
many different points and plot the results. For instance, at the point , we
have . Using software to graph both the surface and this point gives the
Generating curves on surfaces
Recall that we described curves in by giving vector-valued functions , where the
coordinates of any point on the curve can be determined from a single parameter. We
would now like to consider vector-valued functions alongside functions of
several variables. As usual, we will work with two variables so that we can
better visualize our examples, but our results will also extend to the case of
If we have a function and a vector-valued function so that for any value of , is in
the domain , we can evaluate at each point along and produce another curve on the
Consider again . Also consider the curve defined by in the -plane. Note that the
domain of is all of , so each point on is in the domain of .
Fill out the table below: The points in our table lie on the curve in the -plane. The
points in our table lie on the surface . If we would evaluate for every point on we
would see the curve on the surface corresponding to the curve in the plane, as
We can think of the function as “lifting” a curve onto the surface in -space. Of
course, the curve must be in the domain of the function , and we should always be
cautious when using the notion of “height” for our -coordinate.
So far, we have focused mainly on the curve lying in the domain of a function . Let’s
now focus on the curve on the surface. Since the curve is in , we must use a
parametric equation to describe it. Fortunately, with our background on
vector-valued functions, finding such a description should be straightforward.
Let as before. Give a parametric description of the the curve that lies on this
surface above the line in the -plane.
A parameterization of in the -plane
We can now use the equation of the surface to give in terms of . Since , setting and
expressing in terms of gives . Thus, a parameterization of the curve is
The idea of looking at a curve in the domain of and the corresponding curve on the
surface can be helpful when thinking about many topics that follow. We may see
these ideas again when we discuss limits of functions of several variables, derivatives
and differentiability, the chain rule, tangent planes, as well as constrained
optimization. Any time we work with these curves on surfaces, remember to think
carefully about whether we are working in the domain of or on the surface