Math 15, Winter 2006: Homeworks, Week: 1 2 3 4 5 6 7 8

Homeworks will be due in each Wednesday at the start of class (or to my office beforehand). They will include material covered by the end of class of the Friday prior to the due date . The homework is your best chance to practice the techniques learned during the course and as such is vital if you are to understand everything as fully as possible.
Please note that late homework will not be accepted.

Homeworks must be:

• Written up neatly and carefully on 8.5 x 11 paper.
• Ordered so that the grader can easily find all solutions.
• Stapled if there is more than one sheet.
• Clearly marked with your name.

It is advised that you work in groups when solving the homework problems. However, you should write up the final versions individually to ensure that you understand the concepts involved.

Week 9: Wed 3/9

• IMPS 5: 1, 9, 10, 11, 12, 6
Corrections and Clarifications
• On all questions you may use any method you like provided it is valid.

Week 8: due Wed 3/1

• DGC III : 17 ,
• IMPS 4: 6, 8, 9, 10
Corrections and Clarifications
• DGC III: 17 Hint: consider the vector field curl E + dB/dt. Show that this must be expressible as curl A, for some vector field A. Then show that A must be conservative.

Week 7: due Wed 2/22

• DGC II : 11(b), 12, 13(b),
• IMPS 3: 7, 8, 10
Corrections and Clarifications
• DGC 11.11-13: You can also use a potential function method if you prefer.

Week 6: due Wed 2/15

• Read Chapters 1 and 2 of DGC
• DGC II : 5, 8, 10 (a),(b), 23 (read comments below)
• IMPS 2: 21, 23, 24, 25(b)
Corrections and Clarifications
• DGC sometimes needs to use cumbersome notation and formulas, because they do everything in terms of graphs rather than parametric surfaces X(u,v). Note that equations II-12 and II-13 are the graph forms of the versions given in lecture.
• Gauss's Law (II-1) is a physical interpretation of what we were doing at the end of Wed. (2/8). Do you see how?
• DGC II.5: Using the parametric forms (rather than II-13) would be preferable here.
• DGC II.23: For each part, compute both integrals by deciding which integral is easier and just computing that.

Week 5: due Wed 2/8

• Do the problems in the following pdf file: Homework 5
• Before Wednesday, attempt the practice midterm (appearing soon in the exams section).
Corrections and Clarifications
• Don't hand in your practice midterm solutions.

Week 4: due Wed 2/1

• DGC III (p105-109): 5, 19
• DGC IV (p144): 8
• IMPS 2: Exercises 2, 4, 5, 6
Corrections and Clarifications
• III-5: "nabla X F" is alternative notation for curl(F).
• IV-8: The current density of a fluid is given by J= (rho) v where v is the velocity. You might find it helpful to note curl(uF) = (grad u) x F + u curl F
• 2.6: The unit normal for the flux here will be constant and perpendicular to the plane containing S. The flux integral will now be a two dimensional integral rather than a line integral.

Week 3: due Wed 1/25

• DGC III (p104): 3 (a)-(d), 4 (a)-(b)
• DGC IV (p144): 1 (b) using function (iv) and curve (iii), 4 (b)(i)
• IMPS 1: Exercises 12, 16, 17
Corrections and Clarifications
• IV-1: The question is asking you to compute the integral directly from definition. (i.e. you're not allowed to just quote a theorem)
• IV-4: The non-bold r stands for the vector norm of the bold r.
• 1.12: Recall the electric field is the force the charged object exerts on a charge value +1 at each point. The axis for the disk in this case is the z-axis. (A disk is rotationally symmetric about its axis).
• 1.16: Only compute one way, but you can choose which.

Week 2: due Wed 1/18

• IMPS 1: Read all of Chapter 1, Exercises 4, 5, 8, 9, 10, 11
• IMPS 6: Exercises 14, 17
Corrections and Clarifications
• 1.4: just find the center of mass.
• 1.8: feel free to ignore the hint and use a change of variables if you prefer.
• 1.11: this should read, above the xy-plane.

Week 1: due Wed 1/11

• IMPS 6: Exercises 2, 3, 4, 6, 7, 10, 11, 15, 16
• IMPS 1: Exercises 2, 6 (pages 28, 38)
Corrections and Clarifications
• 6.3: part 1 should be find G o F not F o G.
• 6.4: part 2 its ok to give your answer as a combination of x,y and r,s,t.
• 6.6: should read G(r,"theta") not G(r"theta").
• 6.16: the 3rd component of F should be Hv; here R and H are constants.
• 1.6: use standard x,y,z coordinates for this problem.