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Functions of several
variables
Functions of Several Variables
In this section we want to go over some of the basic ideas about functions of more than one
variable
...
For example here is the graph of z = 2 x 2 + 2 y 2 − 4
...
We saw several of these in
the previous section
...
Another common graph that we’ll be seeing quite a bit in this course is the graph of a plane
...
Recall that the equation of a plane is given by
ax + by + cz = d
or if we solve this for z we can write it in terms of function notation
...
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 instance, the intersection with the z-axis is defined by
x = y = 0
...
Now, to extend this out, graphs of functions of the form w = f ( x, y, z ) would be four
dimensional surfaces
...
We next want to talk about the domains of functions of more than one variable
...
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
...
Example 1 Determine the domain of each of the following
...
[Return to Problems]
(b) This function is different from the function in the previous part
...
There is one for each square root in the
function
...
[Return to Problems]
(c) In this final part we know that we can’t take the logarithm of a negative number or zero
...
Here is a
sketch of this region
...
Example 2 Determine the domain of the following function,
1
f ( x, y , z ) =
x 2 + y 2 + z 2 − 16
Solution
In this case we have to deal with the square root and division by zero issues
...
The next topic that we should look at is that of level curves or contour curves
...
So the equations of the level curves are f ( x, y ) = k
...
You’ve probably seen level curves (or contour curves, whatever you want to call them) before
...
Of course, we probably
don’t have the function that gives the elevation, but we can at least graph the contour curves
...
Example 3 Identify the level curves of f ( x, y ) = x + y
...
2
2
Solution
First, for the sake of practice, let’s identify what this surface given by f ( x, y ) is
...
So, we have a cone, or at least a portion of a cone
...
Note that this was not required for this problem
...
Now on to the real problem
...
In the case of our example this is,
k = x2 + y2
⇒
x2 + y 2 = k 2
where k is any number
...
We can graph these in one of two ways
...
Here is each graph for some values of k
...
The contour will represent the intersection of the surface and
the plane
...
The equations
of level surfaces are given by f ( x, y, z ) = k where k is any number
...
In some ways these are similar to contours
...
Traces of surfaces are curves that represent the intersection of the surface and
the plane given by x = a or y = b
...
Example 4 Sketch the traces of f ( x, y ) = 10 − 4 x 2 − y 2 for the plane x = 1 and y = 2
...
We can get an equation for the trace by plugging x = 1 into the equation
...
2
⇒
z = 6 − y2
Below are two graphs
...
On the right is a graph of the surface and the trace that we are after
in this part
...
Here is the
equation of the trace,
z = f ( x, 2 ) = 10 − 4x − ( 2 )
2
and here are the sketches for this case