Calculus-3
Lecture: Multivariate Functions
Introduction
In this section we introduce functions that have more than one independent variable.
Examples:
Schematically, we have...
NOTE:
In Calculus-1 we had the vertical line test for a function y = f (x) (i.e., each x in
the domain generate only one y in the range. Similarly, we now require that each
in the domain of
the range.
generate only one w in
Example
Consider the ellipsoid...
We can solve the above equation for z to get...
This is a perfectly good relation (between x, y and z), but it is not a function.
NOTICE that:
Two different z-values out for 1 xy-pair going in. NO WAY!!
Evaluating a Multivariate Function
Basically, you just plug in the numbers.
Example
Evaluate the function...
at the point ( 1, 2, -1).
SOLUTION:
We have...
Natural Domain vs. Restricted Domain
Natural Domain
The set of all n-tuples
that can be processed
by the function.
Example
Find and graph the natural domain of the function...
SOLUTION:
We are not allowed to divide by zero so we can use any (x,y) pair except
those for which x - y =0. Thus...
So, the domain is...
The graph shows all the points in the xy-plane
that are part of the domain of the function.
The disallowed points are all on the line y = y.
So, any points but these are in the domain.
The graph of the domain is to the right.
See the following box for graphing
conventions.
Graphing Conventions
A curve drawn as a solid line contains points that are part of the
domain while...
2. A curve drawn with a dashed line contains points that are not
part of the domain.
3. A shaded region of the space can either contain allowed points
or disallowed points, its up to you, but you must indicate which
convention you are using.
1.
Finding the (Natural) Domain of a Function
Many functions have domains that include all real numbers. But some do not. The table below lists a few common functions
whose domains are not so inclusive. Watch for these.
Some Common Functions Whose Domains
Do Not Include All Real X-Values
Function
Domain
Example
Restricted Domain
Any subset of the domain. You can restrict the domain to any
subset of itself that suits your purposes.
Example
We could restrict the domain of the last example
to...
Hence, the restricted domain would be limited to
the points in the box in the figure to the right.
Graphing Mulitvariate Functions
1. Functions having 2 independent variables,
, are just
surfaces. For example...
Is just the paraboloid to the right.
2. Functions having 3 independent variables w = f (x,y,z) require 4
mutually perpendicular axes to draw (one each for x,y&z and one
more for w).. Being objects that live in 4-D space, you can’t sketch them!
3.
The same holds true for functions having more than 4 independent variables. A function
of n independent variables is an object living in (n+1)-D space!
Level Curves
A level curve is just the trace of some plane with a surface that is projected on
one of the coordinate planes (usually the xy-plane).
Example
The figure to the right shows the trace of
the paraboloid...
With the plane...
The associated level curve is also shown.
Notice that it is just the trace translated
vertically downward into the xy-plane.
Traces with different planes generate whole
families of level curves, one curve for each plane. When graphed, these families are
essentially contour maps that show (in the xy-plane) points having equal z-values.
Example
For the paraboloid of the last example...
Sketch the level curves for z = 0, 1, 2, and 3.
SOLUTION:
First we find the trace with a generic plane z = k.
Now, we can graph the circles for k = 0, 1, 2, &
3 in the xy-plane. These are the level curves for
the planes z = 0, 1, 2, & 3.
They are plotted in the figure to the
right. Notice, the closer the spacing
between circles, the steeper the slope on
the paraboloid!
Level Surfaces
A level surface is the collection of points in 3-D space that all have
the same w-value where...
NOTICE:
All xy-points along a level curve have the same z-value on the associated surface.
Here, all points on the level surface have the same w-value on the associated
hypersurface in 4-D space!
Example
For the hypersurface...
Find the level surfaces upon which w = 0, 1, 2.
SOLUTION:
Basically, we are going to set w = 0, then 1, then 2 and see what sort of
3-D surface results.
Let w = k, then we have...
This is just an ellipsoid with all its axes=
surface is a sphere of radius=
, in other words the level
with center @ (0,0,0).
Thus, the level surfaces are...
w = 0 ......... a point @ (0,0,0)
w = 1 ......... A sphere, R= 1, ctr @ (0,0,0)
w = 2 ......... A sphere, R=
ctr @ (0,0,0)