math251-spring2007-lab3_background.html

Maple Lab 3 (Spring 2007 Math 251 Sections 05-07)

Background Worksheet.

It's a good idea to start every sheet with the restart command. Maple remembers everything that

you do, so the restart command gives you a fresh start without having to shutdown and restart

the program itself!

A Quick Note for Typing in Maple:

To create new text groups such as this use the button labeled "T" above. To create a new execution group

(the lines with maple commands which start with ">") you can use the button labeled "[>" or the keyboard

shortcut "Ctrl+J". To insert a group before the cursor use "Ctrl+K".

To delete a line of input or Maple output you can use the keyboard shortcut "Ctrl+Delete". To begin

another line in an execution group use "Shift+Enter". Remember that "Enter" alone will just execute

the command you've typed.

For this lab we will need the "plots" package along with the "VectorCalculus" package.

Warning, the name changecoords has been redefined

Warning, the assigned names `<,>` and `<|>` now have a global binding

Warning, these protected names have been redefined and unprotected: `*`, `+`, `-`, `.`, D, Vector, diff, int, limit, series

We will be exploring Chapter 15 material using Maple. Specifically, we will be looking at:

computing multiple integrals, plotting solids, and finding centroids.

We have all ready seen how to graph functions of two variables: using the command. Now let's

look at how we can use to graph "parametric surfaces". To get started let's look at some 2D examples.

Plot

So we have plotted part of a parabola. Now let's parametrize this curve. Parametrization can

be very tricky in general, but in the case we are looking at a curve of the form: it's

quite simple. Using as our parameter, we simply choose: and .

Now we can graph our piece of the parabola using parametric equations.

Plot

In much the same way, we can parametrize any surface of the form: using

two parameters (say and ) as follows: , , and .

We can plot these surfaces using either in the "old" way or the "new" (parametric) way.

For Example:

Plot

It is also important to note that so far we have been plotting surfaces defined over a rectangle.

In the example above we are graphing over the rectangle: .

Maple can handle plots defined over "type I" or "type II" regions. For example, let's graph

over the region (inside

a circle centered at the origin with radius 4).

The region is both "type I" and "type II", let's consider it as "type II". We find bounds for

in terms of and then constant bounds.

So we get the bounds: and .

Plot

Of course we can do far more with parametric plots. Consider the sphere centered at the origin with radius 5.

In rectangular coordinates it is described by , but in spherical coordinates we have

the much simpler equation "". Using these coordinates, we see that:

, , and where and .

Let's plot our sphere using "parametric plot3d".

Plot

Notice that this plot looks much nicer than what we got in previous labs using !

Let's use this new tool to plot the region of integration for problem #31 in section 15.7.

In this problem we are supposed to rewrite the integral:

as equivalent iterated integrals in the 5 other orders.

We are integrating over the solid region bounded by , , , and .

[It turns out that the other bounds (the upper bounds on the outter two integrals): and

are extraneous. If we plot them along with the bounds listed above, they will just make it harder to

"see" what's going on.]

Let's simultaneously plot all of these surfaces and then make the proper restrictions to get a plot

of our "mystery" solid.

First, the two surfaces: and which make up the "bottom" and "top" of our solid.

Next, the front and back. These surfaces should be of the form: (if we wanted to

plot "sides", we should have surfaces of the form: ). This means we should plot:

(the back) and (the front -- we got this by solving for ).

Plot

From this we can see how to "cut off" irrelevant parts of our surfaces.

Plot

Using this graph, we can better "see" how to write down the remaining 5 iterated integrals.

Let's assume the function is and then compute all 6 iterated integrals.

[If our answers are right, we should get the same result each time.]

Let's "squash out" the -direction and see what region in the -plane with which we are dealing.

Plot

We begin by computing the triple integral in the original order:

In the next two integrals we will place first, so let's see what region of the

-plane we are dealing with when the 's are "squashed out".

Plot

In the final two integrals we will place first, so let's see what region of the

-plane we are dealing with when the 's are "squashed out".

Plot

We have seen how Maple can help us visualize 3D (and 2D) regions of integration.

Now let's make Maple help us compute Jacobians and preform change of variables

for multiple integrals.

Let's make Maple compute the Jacobian for spherical coordinates.

[The following command spits out a "Jacobian matrix" and the "Jacobian determinant".

We just need the latter. Since the command spits out a sequence of answers, we

put a sequence of labels on the left hand side of the assignment (i.e. "").]

(Typesetting:-mprintslash)([M, J := Matrix([[cos(theta)*sin(phi), -rho*sin(theta)*sin(phi), rho*cos(theta)*cos(phi)], [sin(theta)*sin(phi), rho*cos(theta)*sin(phi), rho*sin(theta)*cos(phi)], [cos(phi)...

Don't forget to simplify the Jacobian...

Let's compute the centroid of the "bumpy sphere" which is described by the

equation (in spherical coordinates).

Plot

Looking at the sphere we might guess that the centroid is the origin.

Let's verify this by computing some integrals.

First, the volume of the interior of the bumpy sphere is given by:

where is the interior of the bumpy sphere described in spherical coordinates.

Thus we have...

Next, we must compute the moments about the , , and planes.

For example:

(since and the absolute value of the Jacobian is ).

Then the centroid is: which is...

The origin...just as we suspected!