 |
 |
Multivariable Calculus II (Math 61) Summer 2009 (2nd half) URL: http://www.math.hmc.edu/math61 Section 1 M-F 8:45-10:15 a.m. in Galileo McAlister Section 2 M-F 10:30 a.m.-12:00 p.m. in Galileo McAlister Instructors: Prof. Darryl Yong (dyong@hmc.edu, Olin 1261) Office Hours: 3-4 p.m. and by appointment Tutoring evenings before homework due in Platt Living Room 8-10 p.m. |
The study of functions of more than one independent variable is eminently important, given that we live in a world involving three spatial dimensions (at least). In Math 14, we began learning how to extend the ideas from single-variable calculus to multivariate functions. Along the way, we learned about parametric curves, vector fields, new notions of derivatives (derivative matrices, directional derivatives, gradient, divergence, curl), and new notions of integrals (line integrals and surface integrals).
In Math 61, we will solidify these concepts by generalizing them results to arbitrary coordinate systems and learning about the relationships between all of those important integration theorems. We'll also learn about linearizing multivariate functions (thereby exploring a deep connection between calculus and linear algebra) and multivariate optimization.
Textbooks available from Huntley Bookstore.
Lecture notes will be posted online in the calendar below. These lecture notes will be accessible only to computers on the Claremont Colleges subnet. Mathematica is available for student use at \\charlie.hmc.edu\Dist-Software.
Misc course handouts and useful materials:
| Mon | Tue | Wed | Thu | Fri | |
| Week 4 | June 8 | 9 Recap of Math 14; Curvilinear coordinate systems (Colley 1.7, 5.5) |
10 Jacobian and integration Mathematica notebook |
11 HW 1 due Grad in different coordinate systems; Taylor's approx (4.1) |
12 HW 2 due Linearizing differential equations; Extrema of functions |
| Week 5 | 15 2nd derivative test; Constrained optimization (4.3) Mathematica notebook |
16 HW 3 due Lagrange multipliers |
17 EXAM 1 (outline) |
18 HW 4 due Surface integral defn; Surface integrals on simple surfaces |
19 HW 5 due Parameterized surfaces; and surface integrals over arbitrary surfaces |
| Week 6 | 22 Coordinate-free definition of divergence; divergence in Cartesian and spherical coordinates |
23 HW 6 due Gauss's Theorem and its consequences; Line integrals (review) |
24 HW 7 due Coordinate-free definition of curl; Stoke's Theorem and its consequences |
25 HW 8 due Conservative vector fields and path independence of line integrals |
26 HW 9 due EXAM 2 (take-home) |
|
Exam 1 In class on Wednesday, June 17 |
25% |
|
Exam 2 Take home due Friday, June 26 |
25% |
| Homework | 50% |
No late homeworks will be accepted except for medical or family emergencies. Your lowest homework grade will be dropped. Homework assignments will be announced on this webpage. At the bottom of the table below is a tentative list of all homework problems that will be assigned over the three weeks. You may work ahead if you like, but the problems may change.
You are encouraged to work cooperatively on your homework assignments with your classmates. However, every student MUST write up his/her own homework separately. In addition, you must cite any sources of help that you use. If you work with one of your classmates on a problem, be sure to acknowledge that person in your homework write-up; if you use any software or source besides our textbook, acknowledge that too.
Communicating mathematics well is just as important as being able
to do it. Every assignment should be written up neatly with
explanatory prose where necessary. What is obvious to you may not be
obvious to others, and graders may take points off for writing that
they can't read or understand easily. You may find these guidelines and examples of good and bad mathematical
writing helpful.
| Homework (Due date) |
Assignment |
| HW #1 (Thu 6/11) |
Colley Section 1.7: 40, 42 Colley Section 5.5: 2, 7, 19* Additional problems * Think about how to compute this integral with the least amount of effort. |
| HW #2 (Fri 6/12) |
Colley Section 5.5: 8, 9, 16, 24, 32* (Steinmetz solid picture & Mathematica file) Colley Section 5.6: 22 * Extra credit: Can you also find the surface area of the Steinmetz solid? |
| HW #3 (Tue 6/16) |
Colley Section 4.1: 8, 9, 27 Colley Section 4.2: 1, 4, 6, 7, 8, 32 Colley Section 4.4: 7 |
| HW #4 (Thu 6/18) |
Colley Section 4.3: 1, 8, 16*, 22 (*16c: You may use this graph.) Colley Section 4.6: 1, 7, 20 Additional problem Note: Exam #1 covers material on this problem set. |
| HW #5 (Fri 6/19) |
Surface integral problems |
| HW #6 (Tue 6/23) |
Colley Section 7.1: 1a, 6, 10 Schey II-4c*, II-5ac*, II-10 * You don't have to use Equation II-12 or II-13; use whatever coordinate system makes the problem easiest. |
| HW #7 (Wed 6/24) |
Schey II-8, II-15ab, II-22, II-23bc, II-24, II-25* * Note that the right hand side of this equation is the zero vector, not zero scalar. |
| HW #8 (Thu 6/25) |
Schey III-5, III-12*, III-14**, III-15bc, III-22b*** * This problem makes the most sense in spherical coordinates. ** Note that the right hand side of this equation is the zero vector, not zero scalar. *** The result you obtain in this problem should familiar because it's Green's Theorem! |
| HW #9 (Fri 6/26) |
Schey IV-4a, III-25bac*, III-27** Colley Section 7.6: 17a, 21***, 28 * You do not need to follow III-24 to find H if you can think of a function that works by yourself. ** You only have to show that if G=curl H, then div G=0. You don't have to prove the other direction in this problem. *** When Colley says to computing the answer "directly" in part (a) that means not using any integration theorems. |
| <dyong@hmc.edu> | Last modified: Wed Jun 24 08:39:32 -0700 2009 |