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Mathematical Expression Editor

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We introduce functions that take vectors or points as inputs and output a
number.

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
Earth.

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
topographical map:

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
variables.

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
notation.

The range of is the set of all outputs of . It is a subset of , not .

Consider

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
functions.

In this text, we will use an upper-case letter to denote a function of several
variables.

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 function is defined. We are familiar with
this concept from one-variable calculus, where we would see a function such as and
take its domain to be . In our example , we take its domain to be , as we found
above.

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 young mathematician 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
could have.

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 graph 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 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 height. Also, the thing
we are calling height here is actually the -value of the point on the graph of the
function, and we do not always interpret this as a height. Be careful to keep the
meaning of the function in mind.

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
following.

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
variables.

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
surface.

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
pictured below.

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.

We have already mentioned the
parameterization of in the -plane: We can now use the equation of the
surface , which relates to and , to find 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
itself.

Level sets

It was Descartes who said “Je pense, donc je suis.” He also developed our rectangular
coordinate system, the -plane. This is also known as the Cartesian coordinate
system. This coordinate system allows us to consider the graph of a function.
First, recall that the graph of a function of a single variable, is a curve in a
two-dimensional plane. In the same sense, the graph of a function of two variables,
is a surface in three-dimensional space. The graph of a function of three
variables, is a surface in four-dimensional space. A surfaces in higher than
three dimensions is often called a hypersurface. How can we visualize such
functions? For visualizing functions , a graphing utility like Desmos is really
great. For visualizing functions , GeoGebra is very helpful. However, once
we get to functions (or ), visualizing the graph of the function as we do
in two and three dimensions becomes much more difficult. One powerful
technique to help us understand a function visually is known as sketching level
sets.

Suppose that is a function and is in the range of . A level set corresponding
to is a set in the domain of the function such that for all points in the
set.

When working with functions the level sets are known as level curves.

When we are looking at level curves, we can think about choosing a -value, say , in
the range of the function and ask “at which points can we evaluate the function to
get ?” Those points form our level curve. If we choose a value that was not in the
range of , there would be no points in the -plane for which , and hence no level curve
associated to .

It may be surprising to find that the concept of level sets is familiar to most people,
but they don’t realize it. Topographical maps, like the one below represent the
surface of Earth by indicating points with the same elevation with contour lines.
We also had an example of the contour lines of Meteor Crater as we began this
section.

Another example you may know are isotherms, which are curves along which the
temperature does not change. We see these in weather maps.

Below we see a surface with level curves drawn beneath the surface. Remember
that the level curves are in the domain of the function, not on the surface
itself.

Given that is in the range of , select the statements below that are true.

The level
curves are in the domain of the function.The level curves are in the -plane.The
level curves are in the range of the function.The level curves are on the surface .The level curves can also be thought of as the intersection of the plane with the
surface .

We often mark the function value on the corresponding level set. If we choose
function values which have a constant difference, then level curves are close together
when the function values are changing rapidly, and far apart when the function values
are changing slowly.

Suppose you have a differentiable function with the following set of level curves.
You should interpolate reasonable values of the function between the level curves
which are shown:

Consider the points , , and on the surface . Order the points from least steep to most
steep.

At point the surface is less steep than at point , and the surface is steepest at point .

Since the -values on the level curves are equally spaced in this example, level curves
which are close together indicate more rapid change in the -values, while
level curves which are further apart indicate slower change in the -values.

Now, let’s see if you can identify some simple surfaces based on their level
curves.

Match the following level surfaces to the equations below.

Let’s look at another example.

Suppose that . Sketch the level curves of for , , , , , , and .

First, notice that the
domain of is , and the range of is also . It’s particularly important to notice
that all of the values of we will use to find level curves are in the range of
.

Now let’s find the level curves of for the required values. Each of our level curves will
be of the form Now we just need to substitute all of our values for and plot each of
the following implicit functions:

To make your sketch, either plot these implicit functions with your favorite graphing
device, or recognize that they are crossing lines when and hyperbolas otherwise.
As a gesture of friendship, we have included a graph of these level curves.
Below, we evaluate on our level curves and plot the resulting curves on the surface .

Notice how the difference between consecutive values is always , so we can use the
closeness of the level curves on the -plane to determine how the surface is changing.
Near the level curves of and we can both predict (from our sketch of just the level
curves) as well as see (on our graph of the curves on the surface) that indeed is
growing quickly.

Let’s see another example.

Suppose that . Find the equation of the level curve that passes through in terms of
and , and then find a parametric description for both the level curve as well as the
corresponding curve on the surface.

First, find the equation of the level curve. Note that the level curve consists of all
points in the -plane that give the same value for . Since lies on this curve, and , the
equation of the level curve is , or .

Now, we find a vector-valued function for the level curve, as well as the curve on the
surface. Since the level curve is given by the equation and we can solve for without
too much algebra, we set . Then, . The level curve can be described parametrically
by: The corresponding curve on the surface can be described parametrically
by:

Notice that the -component of the curve on the surface should not require much
calculation since we found the curve on the surface by noting . This means that all of
the -values on the curve on the surface should be .

So far, the level sets we’ve been working with have been curves in . We can easily
generalize to functions . When working with functions , our level sets are also called
level surfaces.

Level sets in higher dimensions

In higher dimensions, we want to try to use what we understand about functions of
one and two variables to try to better understand functions of three or more
variables.

A function of one variable can be visualized as a curve drawn in two
dimensions.

A function of two variables can be visualized as a surface drawn in three
dimensions.

A function of three variables can be visualized as what we will call a
hypersurface drawn in four dimensions.

A function of variables can be visualized as what we will call a
hypersurface drawn in dimensions.

We use the term “hypersurface” to refer to an object which is like a surface, but in
more than three dimensions. Hypersurfaces are difficult to imagine, and can even be
difficult to picture using modern computer utilities.

For a function of three variables, one technique we can use is to graph the level
surfaces, our three-dimensional analogs of level curves in two dimensions. Given ,
the level surface at is the surface in space formed by all points where . It’s time for
an example.

If a point source is radiating energy, the intensity at a given point in space is
inversely proportional to the square of the distance between and . That is, when ,

for some constant . Let ; find the level surfaces of .

First, let’s think about this
situation. If energy (say, in the form of light) is emanating from the origin, its
intensity will be the same all a points equidistant from the origin. That is, at any
point on the surface of a sphere centered at the origin, the intensity should be the
same. Therefore, the level surfaces must be spheres.

We now confirm our thought process mathematically. The level surface at is defined
by Algebra reveals Given an intensity , the level surface is a sphere of radius ,
centered at the origin. Every point on each sphere experiences the same intensity of
the radiating energy.

We have found that the level surfaces of in the above example are concentric spheres.
If we picture several of these concentric spheres at the same time, we can get some
intuition about the graph of in four dimensions in the same way that a collection of
level curves in two dimensions gave us some intuition about the corresponding surface
in three dimensions.

From explicit surfaces to level surfaces

We turn our attention to an important concept that will arise again in future
sections.

Suppose that is a function of two variables. Then, the surface is a
level surface of the function

In fact, if is a function of variables, we can also consider it to be a particular level
set for some other function of variables. This idea is very powerful, as it allows us to
consider the same function from two different perspectives. Having multiple
perspectives gives us extra tools to use when considering our function, as well as
allows us to look at the function in whatever manner we find most convenient. Let’s
consider a specific example.

Suppose that . Find a function such that is a level surface of .

We can move to the
right-hand side to obtain: We can now set and recognize that our original surface is
the level surface of corresponding to .

Again, it appears that all we did here was some easy algebra. We made a new
function of one more variable by simply rearranging the original equation that
defined our surface. But having multiple perspectives is always better than having
only one. In addition to its other uses, the content of this procedure is vital for

finding normal vectors for explicitly defined surfaces.

finding tangent planes for explicitly defined surfaces.

These results will be explored further in later sections. It’s good to become familiar
with these ideas now, so that we can make expert use of them later.