Spring 2006

# Mathematics Math21a Spring 2006

## Multivariable Calculus

Office: SciCtr 434
Email: knill@math.harvard.edu
Harvard Mathematics

 Weekly checklists [PDF]: "Somewhere, over the rainbow ..." Week1 Week2 Week3 Week4 Week5 Week6 Week7 Week8 Week9 Week10 Week11 Week12

## Homework

All sections have the same homework during weeks 1-10. Biochem homework will be different in weeks 11-12.
• Homework is due at the beginning of class. The last homework in the semester will be due Tuesday morning 11 AM latest in the mailbox of the CA.
• No late homework accepted but will delete the least 3 scores for MWF sections and 2 for TTh sections. Reserve these 3 "Joker" cards for emergencies.
• 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: 8,10,14,18,34 Due Wed Feb 8 (rsp Thu Feb 9 for TTh) 2. Class 9.2: 18,26,38 9.3: 34,38 Due Fri Feb 10 (rsp Tue Feb 14 for TTh) 3. Class 9.4: 6,14,16,30,34 Due Mon Feb 13 (rsp Tue Feb 14 for TTh) Solutions ```
 ```Week 2: 1. Class 9.5: 8,30,36,48,54 Due Wed Feb 15 (rsp Thu Feb 16 for TTh) 2. Class 9.6: 2,4,10,18,24 Due Fri Feb 17 (rsp Tue Feb 21 for TTh) 3. Class 9.6: 12,22,34 Due Wed Feb 22 (rsp Tue Feb 21 for TTh) Problem A: The global positioning system GPS is based on the fact that a receiver can easily find the difference of distances to two satellites (each GPS satellite sends periodically signals which are triggered by an atomic clock. While the distance to each satellite is not known, the difference from the distances to two satellites can be determined from the time delay of the two signals. The receiver does not need an atomic clock). Given two satellites P=(2,0,0), Q= (0,0,0) in space. Identity the quadric, whose distance to P is by 1 larger than the distance to Q. How many GPS satellites does your GPS receiver have to "see" in space so that it can determine the position? Problem B: Find the surface whose points have the property that the distance to the x-axes is half the distance to the parametrized line r(t) = (1,0,0) + t (0,0,1). Solutions , Problem A, Problem B. ```
 ```Week 3: 1. Class 10.1: 16,32,38 10.2: 28,46 Due Fri Feb 24 (rsp Thu Feb 23 for TTh) 2. Class 10.3: 4,14,18 10.4: 24,38 Due Mon Feb 27 (rsp Tue Feb 28 for TTh) ``` Solutions ,
 ```Week 4: 1. Class 9.7: 10,16,20,28,36 Due Wed Mar 1 (rsp Thu Mar 2 for TTh) 2. Class 10.5: 4,18,28,30,32 Due Fri Mar 3 (rsp Tue Mar 7 for TTh) 3. Class 11.1: 16,22,31-36 11.2: 30,36 Due Mon Mar 6 (rsp Tue Mar 7 for TTh) Solutions ```
 ```Week 5: 1. Class: 11.3:64,66,68,80 (d optional) Due Wed Mar 8 (rsp Thu Mar 9) A: Verify that f(x,t)=e(-r t) sin(x+ct) satisfies the advection equation ft(x,t) = c fx(x,t) - r f(x,t). 2. Class: 11.4: 4,6,26,30,32 Due Fri Mar 10 (rsp Tue Mar 14) 3. Class: 11.5: 4,26,28,32,42 Due Mon Mar 13 (rsp Tue Mar 14) Solutions ```
 ```Week 6: 1. Class: no homework (exam) 2. Class: 11.6: 24,42,44,46, A A: r(u,v) = (u,v3,u3+v2) is a parametrized surface S. a) Find an implicit equation g(x,y,z)=0 for this surface. b) Use a) to find the tangent plane at the point (1,1,2). c) Why are the vectors ru(u,v) and rv(u,v) tangent to S? d) Use c) to find the tangent plane at the point (1,1,2) again. Due Friday, Mar 17 rsp. Tue Mar 21 3. Class: 11.6: 10,26,28,30,36 Due Monday, Mar 20 rsp. Tue Mar 21 Solutions ```
 ```Week 7: 1. Class: 11.7: 2,12,44,48,50 Due Wednesday Mar 22 rsp Thursday Mar 23 2. Class: 11.8: 4,6,10,16,38 Due Friday Mar 24 rsp Thuesday Apr 4 3. Class: 11.8: 18, 24, review page 827: nr 8 Due Monday Apr 3 rsp Tuesday Apr 4 A: Find the extrema of the function f(x,y) = (x2+2y2 ) exp(-x2 -y2) on the circle g(x,y)=x2+y2=4 using the method of Lagrange multipliers. B: Find the absolute maximum of the same function f inside { x2+y2 less or equal to 4 }. ``` Solutions
 ```Week 8: 1. Class: 12.1: 8, 12.2: 8,10,14,32 Due Wed Apr 5 rsp Thu Apr 6 2. Class: 12.3: 4,26,36,42,44 Due Fri Apr 7 rsp Tue Apr 11 2. Class: 12.4: 8,20,24,28,30 Due Mon Apr 10 rsp Tue Apr 11 Solutions ```
 ```Week 9: 1. Class: 12.5: 22, 12.6: 14,24,26,28 Due Wed Apr 12 rsp Thu Apr 13 2. Class: 12.7: 4,12,32,44,48 Due Fri Apr 14 rsp Tue Apr 15 3. Class: 12.8: 4,8,16,32,36 Due Mon Apr 17 rsp Tue Apr 15 Solutions ```
 ```Week 10: 1. Class: no homework (exam) 2. Class: 13.1: 10,11-14,15-18,24,34 Due Fri Apr 21 rsp Tue Apr 22 3. Class: 13.2: 12,18,20,34,42 Due Mon Apr 24 rsp Tue Apr 22 Solutions ``` ```Bio chem section: 2. Class: p.10-12 #1,4,6,7 p.22-24 #5,9,14 Due Fri Apr 21 3. Class: p.22 #3,12,17, p.34 #1,4,6 Due Mon Apr 24 Solutions ```
 ```Week 11: 1. Class: 13.3: 8,16,22,26,32 Due Wed Apr 28 rsp Thu Apr 29 2. Class: 13.4: 6,8,12,14,18 Due Fri Apr 31 rsp Tue May 2 3. Class: 13.5: 6,10,16,27,36 Due Mon May 1 rsp Tue May 2 ``` Solutions ```Bio chem section: 1. Class p. 35: 3, 5, 14 Due Wed p. 50: 1, 4 2. Class p. 50: 7, 9, 11, 12, 14 Due Fri 3. Class p. 50: 5, 16, 17, 20 Due Mon ``` Solutions
 ```Week 12: 1. Class: 13.6: 20,22,26,40,42 2. Class: 13.7: 4,8,10,14,18 3. Class: 13.8: 2,10,26 13.9: 34,38 Due Tue in reading period Solutions 12.8 Solutions 12.9 ``` ```Bio chem section: 1. Class: p.65 #2, 3, 8, 13 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 refers 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 does not know where the prize is but picked a random door which happened to have a goat beind it? ``` 2. Class: (1) Three cards are chosen from three different decks. What is the probability of getting at least one spade? (2) In a poker hand, John has a very strong hand and bets 5 dollars. The probability that Mary has a better hand is .04. If Mary had a better hand she would raise with probability .9, but with a poorer hand she would only raise with probability .1. If Mary raises, what is the probability that she has a better hand than John does? (3) When playing bridge, what is the probability that each of the four players receives an ace? Why is the following argument wrong? "It does not matter where the first ace goes. The second ace must go to one of the other three players and this occurs with probability 3/4. Then the next must go to one of two, an event of probability 1/2, and finally the last ace must go to the player who does not have an ace. This occurs with probability 1/4. The probability that all these events occur is the product (3/4)(1/2)(1/4) = 3/32." (4) Let U, V be random numbers chosen independently from the interval [0, 1]. Find the probabity density, distribution function, expected value and variance of the following random variables. (a) Y = max(U, V ). (b) Y = min(U, V ). 3. Class Problems PDF Solutions