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

 Mainpage Syllabus Calendar Homework Exams Supplements Comput. assigns Links

 Suggested extra problems for regular sessions
 Book section Suggested problems 1. Week 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.
 Book section Suggested problems 2. Week 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.
 Book section Suggested problems 3. Week 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.
 Book section Suggested problems 4. Week 10/12 Sections 2.3-2.4: pgs 134-136 # 5, 7.
 Book section Suggested problems 5. Week 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
 Book section Suggested problems 6. Week 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).
 Book section Suggested problems 7. Week 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.
 Book section Suggested problems 8. Week 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)
 Book section Suggested problems 9. Week Section 5.5 & Handout on Surface Area: pgs 286-287 # 1, 3.
 Book section Suggested problems 10. Week 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: 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: F = (y2 ­ 2x, x + z, cos(y) + z). F = (sin(z ­ y2), y2 + 2y + z4, x4 ­ 1). 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. 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. 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. 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. 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. Write down a vector field whose components are not constant, but that has zero curl and zero divergence. 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. 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. Explain why Green's theorem is a special case of Stokes' theorem.
 Book section Suggested problems 12. Week Handout on DEq's, Sections 1-3 a-c: Suggested problems, Answers
 Book section 13. Week No homework.

 Suggested extra problems for Physics sections
 Book section Suggested problems 1. Week 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: 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? 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?
 Book section Suggested problems 2. Week 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: 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: 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.)
 Book section Suggested problems 3. Week 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: 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.
 Book section Suggested problems 4. Week 10/12 Section 2.3: pgs 115-119 # 15, 19. pgs 124-127 # 1, 3, 9, 17, 15, 19.
 Book section Suggested problems 5. Week 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)
 Book section Suggested problems 6. Week 10/22 Section 2.7: pgs 156-159 # 3, 11, 13, 19. Also do: 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
 Book section Suggested problems 7. Week 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. 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.
 Book section Suggested problems 8. Week Section 3.4 & Supplement on Center of Mass: pgs 207-209 # 1, 5, 7,9,11,15. Also do 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).
 Book section Suggested problems 9. Week Sections 5.4 pgs 279-280 # 1 (no technology).
 Book section Suggested problems 10. Week Section 5.5, Handout on Surface Area, & Supplement on Charge Density: pgs 286-287 # 1, 3. Also do: 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. 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.
 Book section Suggested problems 11. Week (The solutions to the additional problems are at the bottom of this page) Section 5.7: pgs 300-301 # 1, 7, 11. Also consider: 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: F = (y2 ­ 2x, x + z, cos(y) + z). F = (sin(z ­ y2), y2 + 2y + z4, x4 ­ 1). 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. 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. 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. 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 C¹s radius. 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 C¹s radius. Write down a vector field whose components are not constant, but that has zero curl and zero divergence. 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. 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. Explain why Green's theorem is a special case of Stokes' theorem.
 Book section Suggested problems 12. Week Handout on DEq's Suggested problems, Answers
 Book section 13. Week No.

 Suggested extra problems for BioChem sections
 Book section Suggested problems 1. Week 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.
 Book section Suggested problems 2. Week 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.
 Book section Suggested problems 3. Week 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.
 Book section Suggested problems 4. Week 10/12 Section 2.3: pgs 115-119 # 15, 19. pgs 124-127 #1,3,9,15,17,19
 Book section Suggested problems 5. Week 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
 Book section Suggested problems 6. Week 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).
 Book section Suggested problems 7. Week 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.
 Book section Suggested problems 8. Week 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.
 Book section Suggested problems 9. Week Rosner 3.7: pgs 68-73 # 3.29, 3.30, 3.74, 3.75, 3.96, 3.97.
 Book section Suggested problems 10. Week Rosner 4.1-4.8: pgs 108-110 # 4.1-4.4, 4.8, 4.34, 4.35, 4.39-4.43.
 Book section Suggested problems 11. Week 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.
 Book section Suggested problems 12. Week Handout on DEq's, Sections 1-3a-c: Suggested problems, Answers
 Book section 13. Week No homework.

 Suggested extra problems for computer science sessions
 Book section Suggested problems 1. Week 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.
 Book section Suggested problems 2. Week 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.
 Book section Suggested problems 3. Week 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.
 Book section Suggested problems 4. Week 10/12 Sections 2.3-2.4: pgs 134-136 # 5, 7.
 Book section Suggested problems 5. Week 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
 Book section Suggested problems 6. Week 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).
 Book section Suggested problems 7. Week 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.
 Book section Suggested problems 12. Week Handout on DEq's, Sections 1-3 a-c: Suggested problems, Answers
 Book section 13. Week No homework.

 Answers to selected problems in the Handout on 3-variable Lagrange multipliers: