math251-spring2007-lab4_background.html

Maple Lab 4 (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 16 material using Maple. Specifically, we will be looking at:

computing "div" and "curl", finding potential functions, computing line integrals, and graphing orientations.

First we define the vector field and test to see if it's conservative

by computing . The curl of will be the zero vector field if it's conservative.

[Notice that we set the coordinates to be "cartesian" coordinates. These are our standard rectangular coordinates.

Maple can also handle other coordinate systems. But we will stick to cartesian coordinates in this lab.]

Since is not zero, we conclude that is not conservative. However, the divergence

of is zero.

Let's try another vector field. Let

This time . Let's find a potential function for .

Consider the line integral where is the upper-half of the circle lying in the xy-plane

centered at (2,0,0) with radius 2 oriented counter-clockwise. Let's compute this integral three

different ways.

First, we compute it directly after parametrizing as follows:

, , and

where .

We plug our parametrization into our vector field using the (evaluate vector field

command) and dot the result with the derivative of our parametrization . Then we must integrate

from to (half way around the circle).

Remember that our vector field is conservative. Let's use the fundamental theorem of line

integrals to find the answer.

We all ready have a potential function. We also need to find the start and end points of

our curve .

The start is (4,0,0) and the end is (0,0,0).

Now let's use compute the integral using the relation:

We need to compute both (the unit tangent for ) and . Recall that

We deal with two different types of surfaces in this class.

1) Level surfaces defined by where is some fixed constant.

2) Parametrized surfaces defined by where .

An orientation for a surface is a smooth choice of unit normal vectors -- that is an orientation

is a vector field on a surface such that at each point the vector field describes a (unit length) normal

vector for the tangent plane of the surface.

We know that gives us normal vectors for the tangent planes of a level surface .

So (as long as ≠ 0 ) we get the orientations: and for our surface .

If our surface is parametrized by we can find tangent

vectors for our surface by computing and .

Thus we must have that gives normal vectors for the tangent planes of our surface.

So (as long as ) we get the orientations: n = Typesetting:-delayCrossProduct(r[u], r[v])/abs(Typesetting:-delayCrossProduct(r[u], r[v])) and n = -

Consider the surface defined by where .

We can view as the level surface 0 where . Thus we

get the orientations:

"Downward" orientation:

"Upward" orientation:

We could also view as parametrized by:

, ,

where .

This gives us the following orientations:

"Upward" orientation:

"Downward" orientation:

Let's graph along with it's downward orientation.

[You may ask, "What are those commands and all those complicated formulas for?" Well,

is the sequence command. This allows us to plug in a sequence of values into a formula.

The whole idea of those commands is to list off a bunch of sample normal vectors for our

surface without having to list them individually.]

> $plotn := seq(seq(arrow(subs({u = a, v = b*sqrt(4-a^2)/(4.01-a^2)}, r), subs({u = a, v = b*sqrt(4-a^2)/(4.01-a^2)}, -n), shape = arrow, color = black), b = -4+a^2 .. 4-a^2), a = -2 .. 2); -1$
$plotn := seq(seq(arrow(subs({u = a, v = b*sqrt(4-a^2)/(4.01-a^2)}, r), subs({u = a, v = b*sqrt(4-a^2)/(4.01-a^2)}, -n), shape = arrow, color = black), b = -4+a^2 .. 4-a^2), a = -2 .. 2); -1$
$plotn := seq(seq(arrow(subs({u = a, v = b*sqrt(4-a^2)/(4.01-a^2)}, r), subs({u = a, v = b*sqrt(4-a^2)/(4.01-a^2)}, -n), shape = arrow, color = black), b = -4+a^2 .. 4-a^2), a = -2 .. 2); -1$

Plot

Here is an ellipsoid with it's outward orientation.

Plot