ANNOUNCEMENTS
MTH 255H - Fall 2000

11/29/00

There will be an evaluation tomorrow, Thursday 11/30, at 9:30 AM, in StAg 107 (our regular classroom).

Class will meet as usual when the evaluation ends at roughly 10 AM.

11/27/00

Homework due Thursday, 11/30/00:

1. Find the flux of F = -y i + x j + 3z k up through the hemisphere z = (16-x²-y²)^1/2.

2. Use Stokes' Theorem to find the work done by F = x i + y j + (x²+y²) k around the boundary (oriented couterclockwise as seen from above) of the part of the paraboloid z = 1-x²-y² which lies in the first octant.

3. Use the Divergence Theorem to calculate the flux of F = x³ i + 2xz² j + 3y²z k outwards across the surface of the solid bounded by the paraboloid z = 4-x²-y² and the xy-plane.

11/22/00

You can find a sample final (not written by me) here.

(The solutions are here, but please try the problems without peeking first.)

11/20/00

A Java-based "microscope" for visualizing vector fields can be found here.

This tool allows one to see the divergence and curl directly; try it!

11/6/00

Homework due Thursday, 11/9/00:

1. Is there a vector field G such that curl G = xy² i + yz² j + zx² k? Explain.

2. Verify the following identities: (Optional: Use spherical coordinates when helpful.)

(a) div r = 3

(b) div (r r) = 4r

(c) div (grad r³) = 12r

Extra Credit: Can you extend the argument in problem 2 so as to give formulas for the divergence and curl of any radial vector field, that is, any vector field of the form F = f(r,theta,phi) rhat?

(You can check your answer in the optional text by Schey or in most introductory texts on electromagnetism.)

Note: r is the position vector in 3-d; r is the spherical radial coordinate. An analogous problem can be done with r the position vector in 2-d and r the polar radial coordinate, but then the answers change, to 2, 3r, and 9r, respectively.

11/1/00

The Murder Mystery Method consists of taking the indefinite integrals of each of the components of a vector field F with respect to the corresponding variable, and regarding each result as a clue to the potential function f. The key things to remember are:

The clues must be consistent.

Each clue only counts once.

Consistency means that each function of n variables must occur in exactly n of the integrals. This can be used instead of taking the curl to determine whether F is conservative:

F is conservative if and only if the consistency condition is satisfied.

Assuming F is conservative, the potential function is obtained by combining all the clues, being careful to use each term which appears in more than one integral only once.

Always check your answer by taking the gradient of your potential function.

Here's why the Murder Mystery Method works. If the potential function is a function of n variables, with n=1,2,3, then when taking the gradient of f you will get precisely n nonzero derivatives. For instance, if f=xyz, then all 3 partial derivatives are nonzero, but if f=xy only the x and y partial derivatives are nonzero. In the Murder Mystery Method, all you're doing is integrating these partial derivatives, so of course you get back what you started with. But you get it for each integral! Thus, any part of f involving n variables must appear in precisely n integrals.

You can practice this method on Problems 13-18 in §16.5.

10/30/00

There will be an in-class midterm this week. Ground rules are:

closed book

no calculators

one handwritten 8½×11 sheet of paper (one side only) with notes

The material covered corresponds to Lessons 1-10 in the Study Guide.

You can find sample exams here and here; these exams were not written by me, nor were they intended for honors students. Answers to the first exam can be found here; solutions to the second exam (in PDF format) can be found here.

The exam will begin Thursday in class. At the end of the hour, you will turn in everything, including the exam and your notes. Between Thursday and Friday's class periods, you may discuss the exam with each other, but not with anybody else. You may also use any written reference materials you like. But you may not bring any written notes with you on Friday.

10/29/00

Two people are talking on the telephone. Both are in the continential United States. One is in a state which borders the Atlantic Ocean; the other is in a state which borders the Pacific Ocean. They suddenly realize that the correct local time in both locations is the same!

10/27/00

Here's a good example of when to use the "long" version of the area corollary. Consider the graph of the equation x^2/3+y^2/3=1. (Try to graph this on your calculator or computer!) A paramaterization is x=cos³t; y=sin³t, with t going from 0 to 2*Pi. Find the area inside using Green's Theorem. (The correct answer is 3*Pi/8.)

10/20/00

Take a look at the figures shown in the announcements for the other class.

10/11/00

Today's Lagrange multiplier example (f=x²+y², g=x²+xy+y²) is best solved by eliminating lambda rather than using matrices. Please note that the equations I wrote down were incorrect, both in components and in matrix form. (I left out some factors of lambda.) Careful: You'll get the same answer whether or not you correct this mistake!

For what it's worth, the curve g=3 is a tilted ellipse, which can be seen from the formula g = (x-y)²/4 + 3 (x+y)²/4 which in turn leads to the explicit parameterization x=sin(t)+3^1/2cos(t), y=sin(t)-3^1/2cos(t). You are not expected to be able to derive these equations.

10/6/00

Take a look at this week's announcments for the other class, especially the pictures.

9/27/00

If you are using Maple to graph parametric curves, you can use the command "plot" for 2d curves, for example:

plot([cos(t),sin(t),t=0..2*Pi]);

but for 3d parametric curves you must use "spacecurve", for example:

with(plots):
spacecurve([t,t^2,t^3],t=0..1);

Note that the syntax of these 2 commands is different!

9/26/00

Please send me an email message sometime this week.

Please include a sentence or two about what your career goals are, and anything else you'd like me to know.

9/25/00

Please note the corrected section numbers on the suggested problems for Lesson 2.

9/22/00

Make sure to read the note about the various editions of the text.