Math 21 a Math21a, Fall 2000
Course Head: Prof. Clifford H. Taubes
Dini surface: surface of constant negative curvature

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Suggested extra problems for Physics sections
1. Week of 9/25-9/29
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. 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 of 10/2-10/6
Book section Suggested problems
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.
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).
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 of 10/9-10/13
Book section Suggested problems
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.
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?
4. Week of 10/16-10/20
Book section Suggested problems
Section 5.2: pgs 263-264 # 1, 5, 7.
Section 2.3: pgs 115-119 # 15, 19. pgs 124-127 # 1, 3, 9, 17, 15, 19.
5. Week of 10/23-10/27
Book section Suggested problems
Section 2.4: pg 127 # 21. pgs 134-136 # 5, 7, 9.
Appendix C and Supplement on relativity: pg 92 # 16, 18. pgs 395 # 3, 5.
Sections 2.5-2.6: pgs 134-136 # 9, 11, 13. pg 140 # 1a-d. pgs 145-146 # 3 (no technology)
6. Week of 10/30-11/3
Book section Suggested problems
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.
Sections 2.7 & 4.4: pgs 157-159 # 21, 23. pgs 245-248 # 1, 3.
Section 4.4 and Handout on 3-variable Lagrange Multipliers: pgs 247-248 # 7 and problems in the Lagrange Multiplier handout.
7. Week of 11/6-11/10
Book section Suggested problems
Section 2.8: pgs 168-169 # 1, 3, 5a-d, 9.
Section 3.1-3.2: pgs 179-182 # 5a,c, 7, 9, 11. pgs 191-193 # 1, (no technology).
Section 3.2 & Handout on Triple Integrals: pgs 192-193: # 5, 9, 11.
8. Week of 11/13-11/17
Book section Suggested problems
Section 3.3 and Appendix B: pgs 199-200 # 1, 5a, 7. pg 385 # 11, 13, 17.
Section 3.4 & Supplement on Center of Mass: pgs 207-208 # 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.
9. Week of 11/20-11/24
Book section Suggested problems
Sections 5.1-5.3: pgs 273-275 # 1, 5, 7, 9, 11.
10. Week of 11/27-12/1
Book section Suggested problems
Section 5.4: pgs 279-280 # 1 (no technology).
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.
Section 5.6, Curl/Div Handout & Supplement on Electricity/Magnetism: pgs 292-293 # 1, 3, 7, 9.
11. Week of 12/4-12/8
Book section Suggested problems
Section 5.7: pgs 300-301 # 1, 3, 7, 9, 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 Cıs 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 Cıs 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 of 12/11-12/15
Book section Suggested problems
Handout on DEq's To be provided.
13. Week of 12/18-12/22
Book section Suggested problems
No homework.

Last update, 9/20/2000, math21a@fas.harvard.edu