In this section we want to go over some of the basic ideas
about functions of more than one variable.
First, remember that graphs of functions of two variables, are surfaces in three dimensional space. For example here is the graph of .
This is an elliptic parabaloid and is an example of a quadric surface.
We saw several of these in the previous section. We will be seeing quadric surfaces fairly
regularly later on in Calculus III.
Another common graph that we’ll be seeing quite a bit in
this course is the graph of a plane. We
have a convention for graphing planes that will make them a little easier to
graph and hopefully visualize.
Recall that the equation of a
plane is given by
or if we solve this for z
we can write it in terms of function notation.
To graph a plane we will generally find the intersection
points with the three axes and then graph the triangle that connects those three
points. This triangle will be a portion
of the plane and it will give us a fairly decent idea on what the plane itself
should look like. For example let’s
graph the plane given by,
For purposes of graphing this it would probably be easier to
write this as,
Now, each of the intersection points with the three main
coordinate axes is defined by the fact that two of the coordinates are
zero. For instance, the intersection
with the z-axis is defined by . So, the three intersection points are,
Here is the graph of the plane.
Now, to extend this out, graphs of functions of the form would be four dimensional surfaces. Of course we can’t graph them, but it doesn’t
hurt to point this out.
We next want to talk about the domains of functions of more
than one variable. Recall that domains
of functions of a single variable, ,
consisted of all the values of x that
we could plug into the function and get back a real number. Now, if we think about it, this means that
the domain of a function of a single variable is an interval (or intervals) of
values from the number line, or one dimensional space.
The domain of functions of two variables, ,
are regions from two dimensional space and consist of all the coordinate pairs,
that we could plug into the function and get back a real number.
Example 1 Determine
the domain of each of the following.
(a) In this case we know that we can’t take the square root of a
negative number so this means that we must require,
Here is a sketch of the graph of this region.
[Return to Problems]
(b) This function is different from the function in the previous
part. Here we must require that,
and they really do need to be separate inequalities. There is one for each square root in the
function. Here is the sketch of this
[Return to Problems]
(c) In this final part we know that we can’t take the logarithm
of a negative number or zero.
Therefore we need to require that,
and upon rearranging we see that we need to stay interior
to an ellipse for this function. Here
is a sketch of this region.
[Return to Problems]
Note that domains of functions of three variables, ,
will be regions in three dimensional space.
Example 2 Determine
the domain of the following function,
In this case we have to deal with the square root and
division by zero issues. These will
So, the domain for this function is the set of points that
lies completely outside a sphere of radius 4 centered at the origin.
The next topic that we should look at is that of level curves or contour curves. The level
curves of the function are two dimensional curves we get by setting ,
where k is any number. So the equations of the level curves are . Note that sometimes the equation will be in
the form and in these cases the equations of the level
curves are .
You’ve probably seen level curves (or contour curves,
whatever you want to call them) before.
If you’ve ever seen the elevation map for a piece of land, this is
nothing more than the contour curves for the function that gives the elevation
of the land in that area. Of course, we
probably don’t have the function that gives the elevation, but we can at least
graph the contour curves.
Let’s do a quick example of this.
Example 3 Identify
the level curves of . Sketch a few of them.
First, for the sake of practice, let’s identify what this
surface given by is.
To do this let’s rewrite it as,
Now, this equation is not listed in the Quadric Surfaces section, but if we square
both sides we get,
and this is listed in that section. So, we have a cone, or at least a portion
of a cone. Since we know that square
roots will only return positive numbers, it looks like we’ve only got the
upper half of a cone.
Note that this was not required for this problem. It was done for the practice of identifying
the surface and this may come in handy down the road.
Now on to the real problem. The level curves (or contour curves) for
this surface are given by the equation are found by substituting . In the case of our example this is,
where k is any
number. So, in this case, the level
curves are circles of radius k with
center at the origin.
We can graph these in one of two ways. We can either graph them on the surface
itself or we can graph them in a two dimensional axis system. Here is each graph for some values of k.
Note that we can think of contours in terms of the
intersection of the surface that is given by and the plane . The contour will represent the intersection
of the surface and the plane.
For functions of the form we will occasionally look at level surfaces. The equations of level surfaces are given by where k
is any number.
The final topic in this section is that of traces.
In some ways these are similar to contours. As noted above we can think of contours as
the intersection of the surface given by and the plane . Traces of surfaces are curves that represent
the intersection of the surface and the plane given by or .
Let’s take a quick look at an example of traces.