Math 21 a Math21a, Fall 2001
Course Head: Prof. Clifford H. Taubes

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Suggested extra problems for regular sessions
1. Week
Book section Suggested problems
9/17 Section 1.1: pgs 12-15 # 1, 7, 9, 11, 15, 21, 25.
9/19 Section 1.2 and Appendix A pgs 25-27 # 1, 5, 7, 15. pgs 375-377 #13,15, 17, 19, 21, 23, 61.
9/21 Section 1.3-1.4: pgs 35-37 # 1, 3, 5, 7, 9. pg 46 # 1.
2. Week
Book section Suggested problems
9/24 Section 1.4-1.5: pg 46 # 3, pgs 55-58 # 5, 7, 9, 17a-c & e, 19, 21.
9/26 Sections 1.6-1.7: pgs 69-74 # 3, 7, 9, 11, 13, 17, 25, 31. pg 82 # 5.
9/28 Sections 1.7-1.8: pgs 82-85 # 1, 7, 9, 21, 29, 33. pgs 90-93 # 1, 5, 9, 13, 15.
3. Week
Book section Suggested problems
10/1 Appendix C: pgs 83-85 # 11, 15, 23. pg 93 # 21, 27, 29. pgs 395 # 1, 3, 5.
10/3 Sections 2.1 & 1.1: pgs 103-106 # 3a-c, 5 (but no technology), 7, 13, 17, 19.
10/5 Sections 2.2-2.3: pgs 115-118 # 1, 11, 13, 15, 19. pg 124-127 # 1, 3, 15, 17, 19.
4. Week
Book section Suggested problems
10/12 Sections 2.3-2.4: pgs 134-136 # 5, 7.
5. Week
Book section Suggested problems
10/15 Sections 2.5-2.6: pgs 136 # 9,11,13 pg 140 1a-d pgs 145-146 #3 (no technology)
10/17 Section 2.7: pgs 156-159 number 3,9,11,13,19.
10/19 Sections 2.7 @ 4.4 : pgs 157-159 number 21,22 pgs 245-248 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 168-169 # 1, 3, 5a-d, 9.
10/26 Section 3.1-3.2: pgs 179-182 # 5a,c, 7, 9, 11. pgs 191-193 # 1 (no technology).
7. Week
Book section Suggested problems
10/29 Section 3.2 & Handout on Triple Integrals pgs 192-193 # 5,9,11
10/31 Section 3.3 & Appendix B pgs 199-200, # 1,5,7, page 385 # 11,13,17
Section 3.4: pgs 207-208 # 1, 5, 7, 9, 11, 15.
8. Week
Book section Suggested problems
Section 5.1: pgs 254-255 # 1 (no technology), 3 (no technology), 5.
Section 5.2: pgs 263-264 # 1, 5, 7.
Section 5.3: pgs 273-275 # 1, 7, 9, 11.
Section 5.4: pgs 279-280 # 1 (no technology)
9. Week
Book section Suggested problems
Section 5.5 & Handout on Surface Area: pgs 286-287 # 1, 3.
10. Week
Book section Suggested problems
Section 5.6 & Handout on Curl and Divergence: pgs 292-293 # 1, 3, 5, 7, 9.
11. Week (The solutions to the additional problems are at the bottom of this page).
Section 5.7: pgs 300-301 # 1, 7, 11
Section 5.7: pgs 300-301 # 3, 9, Also consider:
  1. Use the divergence theorem to compute the flux of the following vector fields outward through the surface of the ball where x2 + y2 + z2 1:
    1. F = (y2 2x, x + z, cos(y) + z).
    2. F = (sin(z y2), y2 + 2y + z4, x4 1).
  2. Suppose that F is a vector field in space with div(F) = 3 and that is, at all points where z = 0, tangent to the xy-plane. Compute the flux of F outward through the z 0 portion of the surface of the ball where x2 + y2 + z2 1.
  3. Write down a vector field with vanishing divergence and with flux equal to through the z 0 portion of the surface of the ball where x2 + y2 + z2 1.
  4. Write down a vector field whose curl is equal to (1, 0, 0). Exhibit such a vector field whose path integral around the circle where x = 0 and y2 + z2 = 1 is equal to 2 ; or else explain why there aren't any.
  5. Exhibit a vector field, F, with the following property: Whenever C is a circle on which y is constant (so parallel to the xz plane), then the path integral of F around the circumference of C is equal to the square of C's radius. Here, orient C counter-clockwise when the axis in the xz plane are drawn so that z is vertical and increasing upwards and x is horizontal and increasing to the right.
  6. Exhibit a vector field, F, with the following property: Whenever C is a circle on which y is constant (so parallel to the xz plane), then the path integral of F around the circumference of C is equals the constant value of y times the square of C's radius. Orient the circle as in Problem 5.
  7. Write down a vector field whose components are not constant, but that has zero curl and zero divergence.
  8. Write down a vector field whose divergence is not everywhere zero, but whose flux through the surface of the ball where x2 + y2 + z2 1 is zero.
  9. Write down a vector field whose path integral is zero around all circles with z = 0 and x2 + y2 = constant, but whose curl has component along (0,0,1) which is not everywhere zero.
  10. Explain why Green's theorem is a special case of Stokes' theorem.
12. Week
Book section Suggested problems
Handout on DEq's, Sections 1-3 a-c: 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 12-15 # 1,7, 9, 11, 15, 21, 25.
9/19 Section 1.2 & Appendix A: pgs 25-27 # 1, 5, 7, 15. pg 375-377 # 13, 15, 17, 19, 21, 23, 61.
9/21 Section 1.3-1.4: pgs 35-37 # 1, 3, 5, 7, 9. pg 46 # 1. Also consider:
  1. A kite string exerts a 12-lb pull (the force F has magnitude 12) on a kite and makes an angle of 45 degrees with the horizontal. What are the horizontal and vertical components of the force vector?
  2. Pull a wagon with a force F of magnitude 10-lbs by a rope which makes an angle of 30 with the horizontal. What are the horizontal and vertical components of the force?
2. Week
Book section Suggested problems
9/24 Section 1.4-1.5: pg 46 # 3, pgs 55-58 # 5, 7, 9, 17a-c & e, 19, 21. Also do: pgs 55-58 # 8, 12.
9/26 Sections 1.6-1.7 & Supplement #1 on Work and Energy. pgs 69-74 # 3, 7, 9, 11, 13, 17, 25, 31. pg 82 # 5. Also do pg 71 # 17 and:
  1. Find the work done by a force F = (3, 0, 0) in moving an object along the segment from the origin in R3 to the point (1, 1, 1).
9/28 Sections 1.7-1.8 & Supplements on Planetary Motion & Torque and Angular Momentum: pgs 82-85 # 1, 7, 9, 21, 29, 33. pgs 90-93 # 1, 5, 9, 13, 15. Also do:
  1. A particle moves with position vector r(t) = (cos(2t), sin(2t)/2, cos(2 t)/2). Show that the particle moves in a plane and find the equation for that plane. (Hint: Compute the angular momentum vector.)
3. Week
Book section Suggested problems
10/1 Sections 1.1 & 2.1-2.2: pgs 103-106 # 3a-c, 5 (but no technology), 7, 13, 17, 19. pgs 115-118 # 1, 19.
10/3 Section 5.1 & Supplement #2 on Work and Energy. pgs 115-118 # 11, 13. pgs 254-255 # 1 (no technology), 3 (no technology), 5. Also do:
  1. Suppose that a particle is moved from the origin to the point (1, 1, 1) along the line segment between them in the presence of the force F = (xy, y - z, 3y). How much work is done?
10/5 Section 5.2: pgs 263-264 # 1, 5, 7.
4. Week
Book section Suggested problems
10/12 Section 2.3: pgs 115-119 # 15, 19. pgs 124-127 # 1, 3, 9, 17, 15, 19.
5. Week
Book section Suggested problems
10/15 Section 2.4: pg 127 # 21. pgs 134-136 # 5, 7, 9.
10/17 Appendix C and Supplement on relativity: pg 92 # 16, 18. pgs 395 # 3, 5.
10/19 Sections 2.5-2.6: pgs 134-136 # 9, 11, 13. pg 140 # 1a-d. pgs 145-146 # 3 (no technology)
6. Week
Book section Suggested problems
10/22 Section 2.7: pgs 156-159 # 3, 11, 13, 19. Also do:
  1. Consider the function of two variables (p,) given by f(p, ) = p2/2 + cos(). Here, p can be any real number and 0 2 . This function arises when studying the mechanics of a harmonic oscillator. Find the minima of this function.
10/24 Section 4.4 and Handout on 3-variable Lagrange Multipliers: pgs 157-159 #21,23, pgs 245-248 #1,3,7 and answered problems in Lagrange Multiplier handout.
10/26 Section 2.8: pgs 168-169 #1,3,5a-d,9
7. Week
Book section Suggested problems
Section 3.1 and 3.2: pgs 179-182 #5ac,7,9,22
Section 3.2 and Handout on Triple integrals pgs 192-193 #5,9,11
Section 3.3 & Appendix B pgs 199-200 #1,5a,7. Page 385, 11,13,17.
  1. Let V be the volume inside the cylinder where 0 z 10 and x2 + y2 1. The density function for the interior of this cylinder is (x,y,z) = (100 - z2) (1 - x2 - y2). Compute the total mass in cylinder, and compute the center of mass.
8. Week
Book section Suggested problems
Section 3.4 & Supplement on Center of Mass: pgs 207-209 # 1, 5, 7,9,11,15. Also do
  1. Let V be the volume inside the cylinder where 0 z 10 and x2 + y2 1. The density function for the interior of this cylinder is (x,y,z) = (100 - z2) (1 - x2 - y2). Compute the total mass in cylinder, and compute the center of mass.
Section 5.1-5.3: pgs 273-275 # 1, 5, 7,9,11
Section 5.4: pgs 279-280 # 1 (no technology).
9. Week
Book section Suggested problems
Sections 5.4 pgs 279-280 # 1 (no technology).
10. Week
Book section Suggested problems
Section 5.5, Handout on Surface Area, & Supplement on Charge Density: pgs 286-287 # 1, 3. Also do:
  1. The charge density on the surface of the infinitely long cylinder, where x2 + y2 1 is given by (x,y,z) = e-|z|. Compute the total charge on the cylinder.
  2. The charge density on the sphere where x2 + y2 + z2 = 1 is given by (x, y, z) = z2. Compute the total charge on the sphere.
11. Week (The solutions to the additional problems are at the bottom of this page)
Book section Suggested problems
Section 5.7: pgs 300-301 # 1, 7, 11. Also consider:
  1. Use the divergence theorem to compute the flux of the following vector fields outward through the surface of the ball where x2 + y2 + z2 1:
    1. F = (y2 2x, x + z, cos(y) + z).
    2. F = (sin(z y2), y2 + 2y + z4, x4 1).
  2. Suppose that F is a vector field in space with div(F) = 3 and that is, at all points where z = 0, tangent to the xy-plane. Compute the flux of F outward through the z 0 portion of the surface of the ball where x2 + y2 + z2 1.
  3. Write down a vector field with vanishing divergence and with flux equal to through the z 0 portion of the surface of the ball, where x2 + y2 + z2 1.
  4. Write down a vector field whose curl is equal to (1, 0, 0). Exhibit such a vector field whose path integral around the circle where x = 0 and y2 + z2 = 1 is equal to 2 ; or else explain why there aren't any.
  5. Exhibit a vector field, F, with the following property: Whenever C is a circle on which y is constant (so parallel to the xz plane), then the path integral of F around the circumference of C is equal to the square of Cs radius.
  6. Exhibit a vector field, F, with the following property: Whenever C is a circle on which y is constant (so parallel to the xz plane), then the path integral of F around the circumference of C is equals the constant value of y times the square of Cs radius.
  7. Write down a vector field whose components are not constant, but that has zero curl and zero divergence.
  8. Write down a vector field whose divergence is not everywhere zero, but whose flux through the surface of the ball where x2 + y2 + z2 1: is zero.
  9. Write down a vector field whose path integral is zero around all circles with z = 0 and x2 + y2 = constant, but whose curl has component along (0, 0, 1) which is not everywhere zero.
  10. Explain why Green's theorem is a special case of Stokes' theorem.
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 12-15 # 1,7, 9, 11, 15, 21, 25.
Section 1.2 & Appendix A: pgs 25-27 # 1, 5, 7, 15. pg 375-377 # 13, 15, 17, 19, 21, 23, 61.
Section 1.3-1.4: pgs 35-37 # 1, 3, 5, 7, 9. pg 46 # 1.
2. Week
Book section Suggested problems
Section 1.4-1.5: pg 46 # 3, pgs 55-58 # 5, 7, 9, 17a-c & e, 19, 21.
Sections 1.6-1.7: pgs 69-74 # 3, 7, 9, 11, 13, 17, 25, 31. pg 82 # 5.
Sections 1.7-1.8: pgs 82-85 # 1, 7, 9, 21, 29, 33. pgs 90-93 # 1, 5, 9, 13, 15.
3. Week
Book section Suggested problems
Appendix C: pgs 83-85 # 11, 15, 23. pg 93 # 21, 27, 29. pgs 395 # 1, 3, 5.
Sections 2.1 & 1.1: pgs 103-106 # 3a-c, 5 (but no technology), 7, 13, 17, 19.
Sections 2.2-2.3: pgs 115-118 # 1, 11, 13, 15, 19. pg 124-127 # 1, 3, 15, 17, 19.
4. Week
Book section Suggested problems
10/12 Section 2.3: pgs 115-119 # 15, 19. pgs 124-127 #1,3,9,15,17,19
5. Week
Book section Suggested problems
10/15 Sections 2.5-2.6: pgs 136 # 9,11,13 pg 140 1a-d pgs 145-146 #3 (no technology)
10/17 Section 2.7: pgs 156-159 number 3,9,11,13,19.
10/19 Sections 2.7 @ 4.4 : pgs 157-159 number 21,22 pgs 245-248 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 168-169 # 1, 3, 5a-d, 9.
10/26 Section 3.1-3.2: pgs 179-182 # 5a,c, 7, 9, 11. pgs 191-193 # 1 (no technology).
7. Week
Book section Suggested problems
10/29 Section 3.2 & Handout on Triple Integrals pgs 192-193 # 5,9,11
10/31 Section 3.3 & Appendix B pgs 199-200, # 1,5,7, page 385 # 11,13,17
Section 3.4: pgs 207-208 # 1, 5, 7, 9, 11, 15.
8. Week
Book section Suggested problems
Rosner Chapter 2: pgs 40-44 # 2.4, 2.5, 2.6, 2.7, 2.11, 2.12, 2.14.
Rosner 3.1-3.5: pgs 66 #3.1-3.11.
Rosner 3.6: pg 69 #3.49, 3.51, 3.57.
9. Week
Book section Suggested problems
Rosner 3.7: pgs 68-73 # 3.29, 3.30, 3.74, 3.75, 3.96, 3.97.
10. Week
Book section Suggested problems
Rosner 4.1-4.8: pgs 108-110 # 4.1-4.4, 4.8, 4.34, 4.35, 4.39-4.43.
11. Week
Book section Suggested problems
Rosner 4.8-4.12: pgs 108-110 # 4.11-4.13, 4.26-4.31.
Rosner 5.1-5.3: pg 147 # 5.1-5.5
Rosner 5.4, 5.5, 5.7, 5.8: pgs 149-151 # 5.35, 5.36, 5.38, 5.60.
12. Week
Book section Suggested problems
Handout on DEq's, Sections 1-3a-c: 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 12-15 # 1, 7, 9, 11, 15, 21, 25.
Section 1.2 and Appendix A pgs 25-27 # 1, 5, 7, 15. pgs 375-377 #13,15, 17, 19, 21, 23, 61.
Section 1.3-1.4: pgs 35-37 # 1, 3, 5, 7, 9. pg 46 # 1.
2. Week
Book section Suggested problems
Section 1.4-1.5: pg 46 # 3, pgs 55-58 # 5, 7, 9, 17a-c & e, 19, 21.
Sections 1.6-1.7: pgs 69-74 # 3, 7, 9, 11, 13, 17, 25, 31. pg 82 # 5.
Sections 1.7-1.8: pgs 82-85 # 1, 7, 9, 21, 29, 33. pgs 90-93 # 1, 5, 9, 13, 15.
3. Week
Book section Suggested problems
Appendix C: pgs 83-85 # 11, 15, 23. pg 93 # 21, 27, 29. pgs 395 # 1, 3, 5.
Sections 2.1 & 1.1: pgs 103-106 # 3a-c, 5 (but no technology), 7, 13, 17, 19.
Sections 2.2-2.3: pgs 115-118 # 1, 11, 13, 15, 19. pg 124-127 # 1, 3, 15, 17, 19.
4. Week
Book section Suggested problems
10/12 Sections 2.3-2.4: pgs 134-136 # 5, 7.
5. Week
Book section Suggested problems
10/15 Sections 2.5-2.6: pgs 136 # 9,11,13 pg 140 1a-d pgs 145-146 #3 (no technology)
10/17 Section 2.7: pgs 156-159 number 3,9,11,13,19.
10/19 Sections 2.7 @ 4.4 : pgs 157-159 number 21,22 pgs 245-248 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 168-169 # 1, 3, 5a-d, 9.
10/26 Section 3.1-3.2: pgs 179-182 # 5a,c, 7, 9, 11. pgs 191-193 # 1 (no technology).
7. Week
Book section Suggested problems
10/29 Section 3.2 & Handout on Triple Integrals pgs 192-193 # 5,9,11
10/31 Section 3.3 & Appendix B pgs 199-200, # 1,5,7, page 385 # 11,13,17
Section 3.4: pgs 207-209 # 1, 5, 7, 9, 11, 15.
12. Week
Book section Suggested problems
Handout on DEq's, Sections 1-3 a-c: Suggested problems, Answers
13. Week
Book section Suggested problems
No homework.


Answers to selected problems in the Handout on 3-variable 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 x0.04 y 0.08 = 104. 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 (r2 + h r) subject to the constraint r2 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 x2/4 + y2/9 + z2/4 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 x2/4 + y2/9 + z2/4 = 1. The maximum occurs where x = 2/, y = , z = 2/.
10) An open-top 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 open-top cylindrical box with radius r and height z has volume r2h and surface area r2 + 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 300x2/5y1/2z1/10 = 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 500x7/10 y1/2 = 40,000. The minimum occurs at x = 32, y = 50.
b) You are being asked to maximize 500x7/10y1/2 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 300x1/2 y1/3 = 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 300x1/2y1/3 subject to 100x + 120y = 25,000. The maximum occurs at x = 150, y = 250/3 and equals 7500 25/6 31/6.
13e) You are asked to evaluate P at x = 150, y = 250/3. The answer is 25 25/6 31/6.
Answers to 11. Week non-text 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 x-y plane at z = 0, its flux is zero through the disk where x2 + y2 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 = (x2, 0, 0) has this property.
9) F = (0, x2, 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 R3. Then, curl(F) = (0, 0, g - f) and so if R is a region in the x-y plane, and C is its boundary curve oriented to be traversed in the counter-clockwise direction, then Stokes theorem says that
C Fdx = 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 non-text book 11. Week Physics section suggested problems

Section 1.3-1.4:
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 1.6-1.7:
1) The work is 3 as measured in the implied units.
Sections 1.7-1.8:
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.


Last update: 11/26/2001, math21a@fas.harvard.edu