Fall 2005

Mathematics Math21a Fall 2005

Multivariable Calculus

Course Head: Oliver Knill
Office: SciCtr 434
Email: knill@math.harvard.edu
Harvard Mathematics

 Weekly checklists: "Somewhere, over the rainbow ..." Week1 Week2 Week3 Week4 Week5 Week6 Week7 Week8 Week9 Week10 Week9 Week10

Homework

All sections have the same homework during weeks 1-10. Biochem homework will be different in weeks 11-12. Here are scanned versions of the problems in Rozanov. If you solve other problems from the book to prepair for the exam, look here.
• Homework is due at the beginning of class.
• Please staple your homework.
• No late homework accepted but will delete the least 3 scores (that is one week of HW for MWF sections and one week of HW for TTh sections). Reserve these 3 "Joker" cards for emergencies.
Students who can not be in class October 2-3 due to Roshe Shana have the option to turn in the homework with their next homework at those days. The same applies for the Yom Kippur holiday, which is Thursday, October 13.
• Collaborations are welcome however you must turn in your own copy and list on it the names of your collaborators.
• Please write legibly and indicate clearly, where a new problem starts.
• Try to keep the problems in order.
• Use words to explain your work, if necessary. Show your work.
• If unable to finish a problem, indicate where you are stuck. This will help us to give partial credit.
• Please infom about typos, misprints. We state here corrections to the in class distributed texts as well as to the posted solutions. Each hint gives you a bonus point. The texts posted here are corrected.
• (+Sol) means solutions have been posted. Solution folder (in case, solutions are not yet linked).

 ```Week 1: 1. Class 9.1: 8,10,14,16,18 Due Wed Sep 28 Sol 2. Class 9.2: 16,22,38 9.3: 34,38 Due Fri Sep 30 Sol 3. Class 9.4: 4,14,16,26,30 Due Mon Oct 3 Sol ``` ```Tuesday/Thursday section (this is a suggestion, the section leader might split differently) 1. Class 9.1: 8,10,14,16,18, 9.2: 16 Due Thu Sep 29 Sol 2. Class 9.2: 22,38 9.3: 34,38 9.4: 4,14,16,26,30 Due Tue Oct 4 Sol ```
 ```Week 2: 1. Class 9.5: 6,32,34,48,54 Due Wed Oct 5 Sol 2. Class 9.6: 2,4,10,14,24 Due Fri Oct 7 Sol 3. Class 9.6: 12,22,34 Due Wed Oct 12 Sol Problem A: What surface does the equation represent? Problem B: Find the diameter of the surface S: . In other words, what is the maximal distance of two points P,Q on S? Solutiosn AB (LaTeX) ``` ```Tuesday/Thursday section (this is a suggestion, the section leader might split differently) 1. Class 9.5: 6,32,34,48,54 Due Thu Oct 6 Sol 9.6: 2,4 2. Class 9.6: 10,14,24 Due Tue Oct 11 Sol 9.6: 12,22,34, Problem A,B Solutiosn AB (LaTeX) ```
 ```Week 3: Due to Columbus day, no class on Monday 1. Class 10.1: 14,34,38 Due Fri Oct 14 Sol 10.2: 28,36 2. Class 10.3: 2,14,18 Due Mon Oct 17 Sol 10.4: 6,24 (more info to problem 10.4.24) ``` ```Tuesday/Thursday section: 1. Class 10.1: 14,34,38 Due Thu Oct 13 Sol 10.2: 28,36 2. Class 10.3: 2,14,18 Due Tue Oct 18 Sol 10.4: 6,24 (more info to problem 10.4.24) ```
 ```Week 4: Monday is review, no homework assigned due to midterm 1. Class 9.7: 10,18,20,32,36 Due Fri Oct 21 Sol 2. Class 10.5: 4,18,28,30,32 Due Mon Oct 24 Sol ``` ```Tuesday: one hour review. The section leader might package the assignments differently: 1. Class 9.7: 10,18,20,32,36 Due Thu Oct 20 Sol 2. Class 10.5: 4,18,28,30,32 Due Tue Oct 25 Sol ```
 ```Week 5: 1. Class: 11.1:22 11.2: 36 11.3: 12,64,66 Due Wed Oct 26 Sol 2. Class: 11.3:68,72,74,80, Due Fri Oct 28 Sol A: Verify that f(x,t)=exp(-r t) sin(x+ct) satisfies the advection equation ft(x,t) = c fx(x,t) - r f(x,t). 3. Class: 11.4: 2,4,26,30,32 Due Mon Oct 31 Sol ``` ```Tue/Thu: (might be split differently in your section) 1. Class: 11.1:22 11.2:36 11.3:12,64,66,68 Due Thu Oct 27 Sol 2. Class 11.3:72,74,80, and Problem A Sol 11.4:2,4,26,30,32 Due Tue Nov 1 ```
 ```Week 6: 1. Class: 11.5: 2,32,28,26,36 Due Wed Nov 2 Sol 2. Class: 11.6: 24,42,44,A,B Due Fri Nov 4 Sol A: r(u,v) = (u,v2,u2+v2) is a parametrized surface S. a) Find an implicit equation g(x,y,z)=0 for this surface. b) Use Aa) to find the tangent plane at the point (1,1,2). B: a) Why are the vectors ru(u,v) and rv(u,v) tangent to the S? b) Use Ba) to find the tangent plane at the point (1,1,2) again. 3. Class: 11.6: 8,26,28,36,46 Due Mon Nov 7 Sol ``` ```Tue/Thu: (might be split differently in your section) 1. Class: 11.5: 2,32,36,26,28 Due Thu Nov 3 Sol 11.6: 24,42 2. Class: 11.6: 44,A,B 11.6: 8,26,28,36,46 Due Tue Nov 8 Sol ```
 ```Week 7: 1. Class: 11.7: 2,10,12,44,48 Due Wed Nov 9 Sol 2. Class: 11.8: 4,6,10,16,18 Due Mon Nov 14 Sol There is no class on Friday due to Veteransday ``` ```Tue/Thu: (Try to do all the problems for Thursday! Nov 15 is the exam. It is important to do those problems before the exam.) 1. Class: 11.7: 2,10,12,44,48 Due Thu Nov 10 Sol 2. Class: 11.8: 4,6,10,16,18 Due Tue Nov 15 Sol ```
 ```Week 8: 1. Class: 12.1: 8, 12.2: 12,16,18,22 Due Fri Nov 18 Sol 2. Class: 12.3: 2,26,36,42,44 Due Mon Nov 21 Sol ``` ```Tue/Thu: 1. Class: 12.1: 8, 12.2: 12,16,18,22 Due Tue Nov 22 Sol 2. Class: 12.3: 2,26,36,42,44 Due Tue Nov 22 Sol ```
 ```Week 9: 1. Class: 12.4: 8,20,24,28,30 Due Wed Nov 23 Sol 2. Class: 12.5: 2,22, 12.6: 2,24,28 Due Mon Nov 28 Sol ``` ```Tue/Thu: 1. Class: 12.4: 8,20,24,28,30 Sol 12.5: 2,22, 12.6: 2,24,28 Due Tue Nov 29 Sol ```
 ```Week 10: 1. Class: 12.7: 4,12,32,44,48 Due Wed Nov 30 Sol 2. Class: 12.8: 4,8,16,32,36 Due Fri Dec 2 Sol 3. Class: 13.1: 24,34 13.2: 18,20,42 Due Mon Dec 5 Sol ``` ```Tue/Thu: 1. Class: 12.7: 4,12,32,44,48 Due Thu Dec 1 Sol 2. Class: 12.8: 4,8,16,32,36 Due Tue Dec 6 Sol 13.1: 24,34 13.2: 18,20,42 ```
 ```Week 11: 1. Class: 13.3: 6,16,22,26,28 Due Wed Dec 7 Sol 2. Class: 13.4: 2,8,12,14,18 Due Fri Dec 9 Sol 3. Class: 13.5: 6,10,14,27,36 Due Mon Dec 12 Sol ``` ```Tue/Thu: 1. Class: 13.3: 6,16,22,26,28 Due Thu Dec 8 Sol 2. Class: 13.4: 2,8,12,14,18 Due Tue Dec 13 Sol 13.5: 6,10,14,27,36 ```
 ```Week 12: 1. Class: 13.6: 20,22,38,40,42 Due Wed Dec 14 Sol 2. Class: 13.7: 2,8,10,14,18 Due Fri Dec 16 Sol 3. Class: 13.8: 2,10,24 13.9: 34,38 Due Mon Dec 19 Sol Problem 13.8:24: replace 2x+2y+z2 by 2x i + 2y j + z2 k ``` ```Tue/Thu: 1. Class: 13.6: 20,22,38,40,42 Due Thu Dec 15 Sol 2. Class: 13.7: 2,8,10,14,18 Due Tue Dec 20 Sol 13.8: 2,10,24 13.9: 34,38 Problem 13.8:24: replace 2x+2y+z2 by 2x i + 2y j + z2 k ```
 ```Week 11 (biochem): 1. Class: Rozanov: 1.1, 1.2, 1.4, 1.10, 2.1, 2.5, 2.6, 2.14 Due Wed Dec 7 rsp. Thu Dec 8 Sol 2. Class: Rozanov: 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.7 Due Fri Dec 9 rsp Tue Dec 13Sol 3. Class: Rozanov: 4.11, 4.12, 4.13, 4.14, 4.15 Due Mon Dec 12 rsp Tue Dec 13Sol ```
 ```Week 12 (biochem): 1. Class: Rozanov: 3.1, 3.2, 3.4, 3.8, 3.13, 3.14, Problem A,B* Due Wed Dec 14 rsp Thu Dec 15 Sol 2. Class: Rozanov: 5.1,5.2,5.3,5.4,5.5,5.7,5.8 Due Fri Dec 16 rsp Tue Dec 20Sol 3. Class: Rozanov: 5.14 and 5.16 Due Mon Dec 19 rsp Tue Dec 20Sol Problem A: Monty Hall Problem: A contestant plays the following game. He or she is presented with three doors. Behind one door is a prize and behind the other two doors are goats. We assume that the contestant prefers the prize to the goat. The contestant picks a door behind which he or she expects to find the prize. Monty Hall, the game show host (who knows where the prize is), opens one of the two other unpicked doors, revealing a goat. The contestant is now given the option of switching his or her choice to the other closed door. (a) Should the contestant switch? (b) What if Monty Hall doesn't know where the prize is but picked a random door which happened to have a goat behind it? Sol. Problem B: Two Envelopes Problem. ((* no need to submit a solution to this problem) John has two envelopes containing X and 2X dollars respectively. He gives one envelope to Jack and one to Jill (which envelope is decided by the flip of a fair coin). Jack opens his envelope and sees Y dollars. He concludes that Jill's envelope contains 2Y or Y/2 dollars with equal probability (since Jack has probability 1/2 of getting the envelope with more money). He calculates that the expected amount of money in Jill's envelope is E(Jill's envelope) = 1/2(2Y + Y/2) = 5Y/4 > Y. So Jack expects Jill's envelope to contain more money. At the same time, Jill performs the same calculation (both are Math21a biochem students) and expects Jack's envelope to contain more money. So they agree to swap envelopes, both expecting to get more money. Is there anything wrong with this argument? Should we expect the total amount of money in the 2 envelopes to increase after swapping? Is this a way of solving the world's financial problems? Sol. ```
 Please send comments to math21a@fas.harvard.edu Background music: Pictures at an Exhibition" by the Russian composer Modest Petrovich Mussorgsky.
 Math21a, Multivariable Calculus, Fall 2005, Department of Mathematics, Faculty of Art and Sciences, Harvard University

Sat Jan 21 22:11:28 EST 2006