Math21a, Fall 2001 Course Head: Prof. Clifford H. Taubes 
Mainpage  Syllabus  Calendar  Homework  Exams  Supplements  Comput. assigns  Links 
Suggested extra problems for regular sessions 
1. Week  
Book section  Suggested problems 

9/17 Section 1.1:  pgs 1215 # 1, 7, 9, 11, 15, 21, 25. 
9/19 Section 1.2 and Appendix A  pgs 2527 # 1, 5, 7, 15. pgs 375377 #13,15, 17, 19, 21, 23, 61. 
9/21 Section 1.31.4:  pgs 3537 # 1, 3, 5, 7, 9. pg 46 # 1. 
2. Week  
Book section  Suggested problems 

9/24 Section 1.41.5:  pg 46 # 3, pgs 5558 # 5, 7, 9, 17ac & e, 19, 21. 
9/26 Sections 1.61.7:  pgs 6974 # 3, 7, 9, 11, 13, 17, 25, 31. pg 82 # 5. 
9/28 Sections 1.71.8:  pgs 8285 # 1, 7, 9, 21, 29, 33. pgs 9093 # 1, 5, 9, 13, 15. 
3. Week  
Book section  Suggested problems 

10/1 Appendix C:  pgs 8385 # 11, 15, 23. pg 93 # 21, 27, 29. pgs 395 # 1, 3, 5. 
10/3 Sections 2.1 & 1.1:  pgs 103106 # 3ac, 5 (but no technology), 7, 13, 17, 19. 
10/5 Sections 2.22.3:  pgs 115118 # 1, 11, 13, 15, 19. pg 124127 # 1, 3, 15, 17, 19. 
4. Week  
Book section  Suggested problems 

10/12 Sections 2.32.4:  pgs 134136 # 5, 7. 
5. Week  
Book section  Suggested problems 

10/15 Sections 2.52.6:  pgs 136 # 9,11,13 pg 140 1ad pgs 145146 #3 (no technology) 
10/17 Section 2.7:  pgs 156159 number 3,9,11,13,19. 
10/19 Sections 2.7 @ 4.4 :  pgs 157159 number 21,22 pgs 245248 number 1,3,5 
6. Week  
Book section  Suggested problems 

10/22 Section 4.4 & Handout on Lagrange Multipliers:  pgs 248 # 7 and problems in the Lagrange Multiplier handout. 
10/24 Section 2.8:  pgs 168169 # 1, 3, 5ad, 9. 
10/26 Section 3.13.2:  pgs 179182 # 5a,c, 7, 9, 11. pgs 191193 # 1 (no technology). 
7. Week  
Book section  Suggested problems 

10/29 Section 3.2 & Handout on Triple Integrals  pgs 192193 # 5,9,11 
10/31 Section 3.3 & Appendix B  pgs 199200, # 1,5,7, page 385 # 11,13,17 
Section 3.4:  pgs 207208 # 1, 5, 7, 9, 11, 15. 
8. Week  
Book section  Suggested problems 

Section 5.1:  pgs 254255 # 1 (no technology), 3 (no technology), 5. 
Section 5.2:  pgs 263264 # 1, 5, 7. 
Section 5.3:  pgs 273275 # 1, 7, 9, 11. 
Section 5.4:  pgs 279280 # 1 (no technology) 
9. Week  
Book section  Suggested problems 

Section 5.5 & Handout on Surface Area:  pgs 286287 # 1, 3. 
10. Week  
Book section  Suggested problems 

Section 5.6 & Handout on Curl and Divergence:  pgs 292293 # 1, 3, 5, 7, 9. 
11. Week (The solutions to the additional problems are at the bottom of this page).  
Section 5.7:  pgs 300301 # 1, 7, 11 
Section 5.7:  pgs 300301 # 3, 9, Also consider: 

12. Week  
Book section  Suggested problems 

Handout on DEq's, Sections 13 ac:  Suggested problems, Answers 
13. Week  
Book section  Suggested problems 

No homework. 
Suggested extra problems for Physics sections 
1. Week  
Book section  Suggested problems 

9/17 Section 1.1:  pgs 1215 # 1,7, 9, 11, 15, 21, 25. 
9/19 Section 1.2 & Appendix A:  pgs 2527 # 1, 5, 7, 15. pg 375377 # 13, 15, 17, 19, 21, 23, 61. 
9/21 Section 1.31.4:  pgs 3537 # 1, 3, 5, 7, 9. pg 46 # 1. Also consider: 

2. Week  
Book section  Suggested problems 

9/24 Section 1.41.5:  pg 46 # 3, pgs 5558 # 5, 7, 9, 17ac & e, 19, 21. Also do: pgs 5558 # 8, 12. 
9/26 Sections 1.61.7 & Supplement #1 on Work and Energy.  pgs 6974 # 3, 7, 9, 11, 13, 17, 25, 31. pg 82 # 5. Also do pg 71 # 17 and: 
 
9/28 Sections 1.71.8 & Supplements on Planetary Motion & Torque and Angular Momentum:  pgs 8285 # 1, 7, 9, 21, 29, 33. pgs 9093 # 1, 5, 9, 13, 15. Also do: 

3. Week  
Book section  Suggested problems 

10/1 Sections 1.1 & 2.12.2:  pgs 103106 # 3ac, 5 (but no technology), 7, 13, 17, 19. pgs 115118 # 1, 19. 
10/3 Section 5.1 & Supplement #2 on Work and Energy.  pgs 115118 # 11, 13. pgs 254255 # 1 (no technology), 3 (no technology), 5. Also do: 


10/5 Section 5.2:  pgs 263264 # 1, 5, 7. 
4. Week  
Book section  Suggested problems 

10/12 Section 2.3:  pgs 115119 # 15, 19. pgs 124127 # 1, 3, 9, 17, 15, 19. 
5. Week  
Book section  Suggested problems 

10/15 Section 2.4:  pg 127 # 21. pgs 134136 # 5, 7, 9. 
10/17 Appendix C and Supplement on relativity:  pg 92 # 16, 18. pgs 395 # 3, 5. 
10/19 Sections 2.52.6:  pgs 134136 # 9, 11, 13. pg 140 # 1ad. pgs 145146 # 3 (no technology) 
6. Week  
Book section  Suggested problems 

10/22 Section 2.7:  pgs 156159 # 3, 11, 13, 19. Also do: 
 
10/24 Section 4.4 and Handout on 3variable Lagrange Multipliers:  pgs 157159 #21,23, pgs 245248 #1,3,7 and answered problems in Lagrange Multiplier handout. 
10/26 Section 2.8:  pgs 168169 #1,3,5ad,9 
7. Week  
Book section  Suggested problems 

Section 3.1 and 3.2:  pgs 179182 #5ac,7,9,22 
Section 3.2 and Handout on Triple integrals  pgs 192193 #5,9,11 
Section 3.3 & Appendix B  pgs 199200 #1,5a,7. Page 385, 11,13,17. 

8. Week  
Book section  Suggested problems 

Section 3.4 & Supplement on Center of Mass:  pgs 207209 # 1, 5, 7,9,11,15. Also do 
 
Section 5.15.3:  pgs 273275 # 1, 5, 7,9,11 
Section 5.4:  pgs 279280 # 1 (no technology). 
9. Week  
Book section  Suggested problems 

Sections 5.4  pgs 279280 # 1 (no technology). 
10. Week  
Book section  Suggested problems 

Section 5.5, Handout on Surface Area, & Supplement on Charge Density:  pgs 286287 # 1, 3. Also do: 

11. Week (The solutions to the additional problems are at the bottom of this page)  
Book section  Suggested problems 

Section 5.7:  pgs 300301 # 1, 7, 11. Also consider: 

12. Week  
Book section  Suggested problems 

Handout on DEq's  Suggested problems, Answers 
13. Week  
Book section  Suggested problems 

No. 
Suggested extra problems for BioChem sections 
1. Week  
Book section  Suggested problems 

Section 1.1:  pgs 1215 # 1,7, 9, 11, 15, 21, 25. 
Section 1.2 & Appendix A:  pgs 2527 # 1, 5, 7, 15. pg 375377 # 13, 15, 17, 19, 21, 23, 61. 
Section 1.31.4:  pgs 3537 # 1, 3, 5, 7, 9. pg 46 # 1. 
2. Week  
Book section  Suggested problems 

Section 1.41.5:  pg 46 # 3, pgs 5558 # 5, 7, 9, 17ac & e, 19, 21. 
Sections 1.61.7:  pgs 6974 # 3, 7, 9, 11, 13, 17, 25, 31. pg 82 # 5. 
Sections 1.71.8:  pgs 8285 # 1, 7, 9, 21, 29, 33. pgs 9093 # 1, 5, 9, 13, 15. 
3. Week  
Book section  Suggested problems 

Appendix C:  pgs 8385 # 11, 15, 23. pg 93 # 21, 27, 29. pgs 395 # 1, 3, 5. 
Sections 2.1 & 1.1:  pgs 103106 # 3ac, 5 (but no technology), 7, 13, 17, 19. 
Sections 2.22.3:  pgs 115118 # 1, 11, 13, 15, 19. pg 124127 # 1, 3, 15, 17, 19. 
4. Week  
Book section  Suggested problems 

10/12 Section 2.3:  pgs 115119 # 15, 19. pgs 124127 #1,3,9,15,17,19 
5. Week  
Book section  Suggested problems 

10/15 Sections 2.52.6:  pgs 136 # 9,11,13 pg 140 1ad pgs 145146 #3 (no technology) 
10/17 Section 2.7:  pgs 156159 number 3,9,11,13,19. 
10/19 Sections 2.7 @ 4.4 :  pgs 157159 number 21,22 pgs 245248 number 1,3,5 
6. Week  
Book section  Suggested problems 

10/22 Section 4.4 & Handout on Lagrange Multipliers:  pgs 248 # 7 and problems in the Lagrange Multiplier handout. 
10/24 Section 2.8:  pgs 168169 # 1, 3, 5ad, 9. 
10/26 Section 3.13.2:  pgs 179182 # 5a,c, 7, 9, 11. pgs 191193 # 1 (no technology). 
7. Week  
Book section  Suggested problems 

10/29 Section 3.2 & Handout on Triple Integrals  pgs 192193 # 5,9,11 
10/31 Section 3.3 & Appendix B  pgs 199200, # 1,5,7, page 385 # 11,13,17 
Section 3.4:  pgs 207208 # 1, 5, 7, 9, 11, 15. 
8. Week  
Book section  Suggested problems 

Rosner Chapter 2:  pgs 4044 # 2.4, 2.5, 2.6, 2.7, 2.11, 2.12, 2.14. 
Rosner 3.13.5:  pgs 66 #3.13.11. 
Rosner 3.6:  pg 69 #3.49, 3.51, 3.57. 
9. Week  
Book section  Suggested problems 

Rosner 3.7:  pgs 6873 # 3.29, 3.30, 3.74, 3.75, 3.96, 3.97. 
10. Week  
Book section  Suggested problems 

Rosner 4.14.8:  pgs 108110 # 4.14.4, 4.8, 4.34, 4.35, 4.394.43. 
11. Week  
Book section  Suggested problems 

Rosner 4.84.12:  pgs 108110 # 4.114.13, 4.264.31. 
Rosner 5.15.3:  pg 147 # 5.15.5 
Rosner 5.4, 5.5, 5.7, 5.8:  pgs 149151 # 5.35, 5.36, 5.38, 5.60. 
12. Week  
Book section  Suggested problems  

Handout on DEq's, Sections 13ac:  Suggested problems, Answers 
13. Week  
Book section  Suggested problems 

No homework. 
Suggested extra problems for computer science sessions 
1. Week  
Book section  Suggested problems 

Section 1.1:  pgs 1215 # 1, 7, 9, 11, 15, 21, 25. 
Section 1.2 and Appendix A  pgs 2527 # 1, 5, 7, 15. pgs 375377 #13,15, 17, 19, 21, 23, 61. 
Section 1.31.4:  pgs 3537 # 1, 3, 5, 7, 9. pg 46 # 1. 
2. Week  
Book section  Suggested problems 

Section 1.41.5:  pg 46 # 3, pgs 5558 # 5, 7, 9, 17ac & e, 19, 21. 
Sections 1.61.7:  pgs 6974 # 3, 7, 9, 11, 13, 17, 25, 31. pg 82 # 5. 
Sections 1.71.8:  pgs 8285 # 1, 7, 9, 21, 29, 33. pgs 9093 # 1, 5, 9, 13, 15. 
3. Week  
Book section  Suggested problems 

Appendix C:  pgs 8385 # 11, 15, 23. pg 93 # 21, 27, 29. pgs 395 # 1, 3, 5. 
Sections 2.1 & 1.1:  pgs 103106 # 3ac, 5 (but no technology), 7, 13, 17, 19. 
Sections 2.22.3:  pgs 115118 # 1, 11, 13, 15, 19. pg 124127 # 1, 3, 15, 17, 19. 
4. Week  
Book section  Suggested problems 

10/12 Sections 2.32.4:  pgs 134136 # 5, 7. 
5. Week  
Book section  Suggested problems 

10/15 Sections 2.52.6:  pgs 136 # 9,11,13 pg 140 1ad pgs 145146 #3 (no technology) 
10/17 Section 2.7:  pgs 156159 number 3,9,11,13,19. 
10/19 Sections 2.7 @ 4.4 :  pgs 157159 number 21,22 pgs 245248 number 1,3,5 
6. Week  
Book section  Suggested problems 

10/22 Section 4.4 & Handout on Lagrange Multipliers:  pgs 248 # 7 and problems in the Lagrange Multiplier handout. 
10/24 Section 2.8:  pgs 168169 # 1, 3, 5ad, 9. 
10/26 Section 3.13.2:  pgs 179182 # 5a,c, 7, 9, 11. pgs 191193 # 1 (no technology). 
7. Week  
Book section  Suggested problems 

10/29 Section 3.2 & Handout on Triple Integrals  pgs 192193 # 5,9,11 
10/31 Section 3.3 & Appendix B  pgs 199200, # 1,5,7, page 385 # 11,13,17 
Section 3.4:  pgs 207209 # 1, 5, 7, 9, 11, 15. 
12. Week  
Book section  Suggested problems 

Handout on DEq's, Sections 13 ac:  Suggested problems, Answers 
13. Week  
Book section  Suggested problems 

No homework. 
Answers to selected problems in the Handout on 3variable Lagrange multipliers: 
5)  The maximum, 14, occurs where x = 7 and y = 0. The minumum, 14, occurs where x = 7 and y = 0. 
6)  The problem is to minimize f(x,y) = 60 x + 100 y subject to the constraint 500 x^{0.04} y ^{0.08} = 10^{4}. The minimum, 2400 (5/6)^{4/5} occurs at x = 20 (5/6)^{4/5} and y = 24 (5/6)^{4/5}. 
7)  Denote the radius of the cylinder by r and the height by h. You are being asked to minimize 2 (r^{2} + h r) subject to the constraint r^{2} h = 50. The minimum occurs where r = (25/)^{1/3} and h = 2 (25/)^{1/3}. 
8)  The point has coordinates (101)^{1/2} (8, 27, 4). One way to find this point is to realize that the gradient of x^{2}/4 + y^{2}/9 + z^{2}/4 is normal to the plane 2x + 3y + z at the points with minimum and maximum distance. 
9)  Suppose that corners of the box have coordinates (±x, ±y, ±z), where x, y and z are positive. These corners will lie on the ellipse (otherwise, the box could be made bigger). Thus, you are asked to maximize the volume, 8xyz, subject to the constraint x^{2/4} + y^{2/9} + z^{2/4} = 1. The maximum occurs where x = 2/, y = , z = 2/. 
10)  An opentop rectangular box of side length x, y, and z (height) has volume xyz and surface area that is equal to xy + 2(xz + yz) = 36. The maximum volume is 3 for x = y = 2 and z = /4. Meanwhile, an opentop cylindrical box with radius r and height z has volume r^{2}h and surface area r^{2} + 2rh = 36. The maximum volume here is 24 (3/)^{1/2} which occurs, when r = h = (12/)^{1/2}. Thus, the cylindrical box has 8 () ^{1/2} times as much volume as the rectangular one. 
11)  You are being asked to minimize the function 80x + 25y + 15z subject to the constraint that 300x^{2/5}y^{1/2}z^{1/10} = 12,000. The minimum occurs for x = 10 (192)^{1/10}, y = 40 (192)^{1/10} and z = 40 (192)^{1/10}/3. 
12a)  You are being asked to minimuze the function 35x + 16y subject to
500x^{7/10} y^{1/2} = 40,000. The minimum occurs at x = 32, y = 50. b) You are being asked to maximize 500x^{7/10}y^{1/2} subject to 35x + 16y = 4800. The maximum occurs at x = 80, y = 125. 
13a)  At ten months, x = 100, y = 125 so the money being spent is 100x + 120y = 25,000 and the production is 300x^{1/2} y^{1/3} = 15,000. 
13b)  You are asked to evaluate P at x = 100 and y = 125. The answer is 75. 
13c)  You are asked to evaluate P at x=100 and y=125. The answer is 40. 
13d)  You are asked to maximize 300x^{1/2}y^{1/3} subject to 100x + 120y = 25,000. The maximum occurs at x = 150, y = 250/3 and equals 7500 2^{5/6} 3^{1/6}. 
13e)  You are asked to evaluate P at x = 150, y = 250/3. The answer is 25 2^{5/6} 3^{1/6}. 
Answers to 11. Week nontext book suggested problems for Section 5.7 
1a)  The divergence of F is 1,
so according to the Divergence Theorem, the flux is equal to 4/3. b) The divergence of F is 2, so the flux is equal to 8/3. 
2)  Since F is tangent to the xy plane at z = 0, its flux is zero through the disk where x^{2} + y^{2} 1 and z = 0. This means that the flux of F through the surface made by joining this disk along its boundary to the boundary of the top half of the ball is equal to the flux of F just through the top half of the ball. With this point understood, the Divergence Theorem asserts that the flux in question is equal to three time the volume of the top half of the ball, thus 2. 
3)  F = (0, 0, 1) has this property. 
4)  F = (0, 0, y) has curl equal to (1, 0, 0). There is no vector field with the given curl and path integral around the circle having absolute value 2. Indeed, according to Stokes theorem, any vector field with curl equal to (1, 0, 0) must have path integral on this circle equal to ±. 
5)  F = (z/, 0, 0) has this property. This can be proved using Stokes theorem. 
6)  F = (zy/, 0, 0) has this property. 
7)  F = (0, z, y) has this property. 
8)  F = (x^{2}, 0, 0) has this property. 
9)  F = (0, x^{2}, 0) has this property. 
10)  Interpret a vector v = (f(x,y), g(x,y)) in the plane as the vector F=(f(x, y),g(x, y),0) in R^{3}.
Then, curl(F) = (0, 0, g  f) and so if R is a region in the xy plane,
and C is its boundary curve oriented to be traversed in the counterclockwise direction, then Stokes theorem says that C F·dx = R curl(F)·k dxdy, where k = (0,0,1). Using the expression just provided for curl(F) turns curl(F)·k into g  f which is the correct integrand for Green's theorem. 
Answers to nontext book 11. Week Physics section suggested problems 
Section 1.31.4:  
1)  If the vertical vector, upward pointing unit vector is denoted by e and the unit vector in the direction of the horizontal projection of F is denoted by e´, then F = 12/ (e + e´). 
2)  Let e denote the unit vector in the direction of the vertical projection of F and let e´ denote the unit vector in that of the horizontal projection. Then, F = 5 (e + e´). 
Sections 1.61.7:  
1)  The work is 3 as measured in the implied units. 
Sections 1.71.8:  
1)  The angular momentum vector is r x r = (/2, 0, 1) which is constant. This means implies that the motion is in the plane where  x/2 + z = 0. 
Section 5.1:  
1)  The work is 5/3 in the implied units. 
Section 2.7:  
1)  The minima is at p = 0 and = , the value there is 1. 
Section 3.4:  
1)  The total mass is 1000/3. The center of mass is (0, 0, 15/4). 
Section 5.5  
1)  The total charge is 4. 
2)  The total charge is 4/3. 