Friday, Jan 5: Line integrals (review), conservative vector fields. Homework due today: none. Related reading: Stewart 17.2, 17.3 (except middle of p. 1096 thru middle of p. 1099) and/or IMPS 10.4, 10.5

Monday, Jan. 8: Double and iterated integrals. Read before class : Stewart 16.1 (skip midpt.rule), 16.2 AND IMPS 14.5.1, 14.5.5 Homework due today: Stewart section 17.2: 18, 21, 38, 40, 41. Section 17.3: 1, 2, 23. (Suggested warm-up problems, not to turn in: 17.2: 17, 37)

Wed. Jan. 10: Lecture 1 (at 8:45 in Wilder) : Iterated integrals over general regions. Lecture 2 (at 10:00 in Bradley) Polar, cylindrical and spherical coordinates. Read before class : Stewart 16.3 and/or or IMPS 14.5.2, 14.5.3, 14.5.4. Also read Steward 11.4 (thru p. 697) and 13.7. Optional IMPS 16.5.3 (thru p. 1109). Homework due today: Stewart section 17.2: 42. Section 16.1: 11, 12, 14. Section 16.2: 7, 14, 28. IMPS p. 927: 458(a). p. 940: 461 (just the first half, i.e., center of mass and moment of inertia about x-axis)

Friday, Jan. 12 : No class today. Please submit homework in manilla envelope by door of 408 Bradley. Homework due today: Stewart section 16.3: 8, 20 (set up iterated integral only; don't evaluate), 27. IMPS pp 937-939: 459 (set up iterated integrals only; don't evaluate), 460 (a-c).

Wed. Jan. 17 : Triple integrals. Integrals in polar coordinates. Read before class : Stewart 16.4 AND your choice of Steward 16.7 or IMPS 14.5.6. Homework due today: Stewart section 11.4: 10, 26. Section 13.7: 10, 14, 16, 18, 20, 28, 38, 40, 60 (also express the region using inequalities involving cylindrical coordinates), 61. Section 16.3: 40. Section 17.2: 43. IMPS text p. 1109: 543 (part I)

Thursday, Jan. 18 (x-hour) Integrals in spherical coordinates. Read before class : Stewart 16.8. Homework due today: none.

Friday, Jan. 19 : Flux (or catch-up if we get behind). Read before class : IMPS 16.1-16.4 Hlomework due today: Stewart 16.4: 11, 18 (set up integral only; don't evaluate), 28, 30 (set up integral only). Section 16.7: 3, 12 (set up integral only), 38 (mass only). Section 17.2: 32 (just find mass. This problem is a review from last term.) IMPS book p. 948-949: 463(d).

Monday, Jan. 22 : flux,. parametrized surfaces. ). Read before class : Stewart 17.6, IMPS 16.4. Homework due today: IMPS 530, 531. Stewart section 17.2: 18 (Ignore the instructions in the text. Instead view the vector field in the picture as a velocity field and determine whether it's flux over each of the two oriented curves shown is positive or negative). Section 13.4: 24 (use cross product), 30. Section 13.5: 30. Section 13.7: 58. Section 15.4: 5 (Also find a vector perpendicular to the tangent plane),

Wed., Jan. 24 : Surfaces integrals. Read before class : Stewart 17.7 and/or IMPS 16.5. Homework due today: IMPS text 535 (I,2). Stewart section 17.6: 2, 3, 12, 14, 17 (explain work), 18, 21, 22, 24, 30 (just find tangent plane. Please try this problem even if we don't go over tangent planes in class Monday.)

Thursday, Jan. 25 (x-hour) Divergence and more discussion of surface integrals as needed. Read before class : IMPS 16.6. Homework due today: none

Fri., Jan. 26 : Divergence Theorem. Read before class : Stewart 17.9 and or IMPS 16.6, 16.7. Homework due today: Stewart section 17.7: 21, 23, 25, 34. IMPS 540 (2), 541..

Mon.., Jan. 29 : Review. Homework due today: IMPS: 545, 551, 553. Stewart Section 17.9: 6, 10, 19. And p. 1155: 39.

Tuesday Jan 30 : Exam. 3:00-6:00 (It's a two hour exam. Please come for any two-hour time period between 3 and 6. ) Bradley 102. Exam covers all material thus far up through and including the Divergence Theorem.

Wednesday, January 31: Proof of the Divergence Theorem, Green's Theorem. Read before class : Stewart 17.4 and review Stewart 17.9. Homework due today: none

Fri., Feb. 2: Conservative vector fields revisited, initial discussion of curl. . Read before class : Stewart 17.3. Homework due today: Stewart section 17.4: 2, 4, 15, 17. Page 1155: 36 (This vector field is familiar. Use what you know from physics.) Also do the following problem: Use Green's Theorem to find the work done by the vector field (x2+y, cos(y2)) in moving a particle along the upper half of the unit circle centered at the origin from (1,0) to (-1,0). (Use a familiar trick.)

Monday, Feb. 5 : Conservative vector fields (continued), surface integrals. Read before class : Stewart 17.6 and /or IMPS 16.5. Homework due today: Stewart Section 17.3: 4, 6, 11, 12, 20, 22, 26, 27. Section 17.4: 3. Section 17.7: 32 (Set up the integral only. Don't evaluate.)

Wed., Feb. 7 : Surface integrals, Volume integrals in spherical coordinates. Read before class : Stewart 16.8. Homework due today: Stewart Section 17.3: 19 (evaluate the line integral in two ways: (1) by using a potential function and (2) by computing the line integral over a conveniently chosen path between the two points (if you choose the path wisely, the computations are very easy). Section 17.6: 33, 34, 35, 36, 40. IMPS text, p. 1116: 548 (3).

 Mon., Feb. 12 : Flux revisited, curl. Read before class : . Homework due today: Stewart section 16.8: 6, 20 (set up integral only), 24, 36. Section 17.7: 4, 8, 10, 14. Stewart p. 1155: 29 (Use Divergence Thm.) IMPS 546, 547.

 Wed., Feb. 14 : Curl, Stokes Theorem. Read before class : Stewart 17.5 and 17.8 and/or hand-out. Homework due today: Stewart 16.8: 23. Section 17.7: 6 (set up integral only), 12 (set up a sum of iterated integrals but don't evaluate), 24 (Do this problem in two ways: first by finding the normal vector field to the surface and setting up and evaluating an appropriate surface integral, and secondly by closing off the region and using the Divergence Thm.). Also do the following problem: Let E be an electrostatic field associated with a charge density given by r(x,y,z)=x2z. Find the flux of E across the sphere x2 + y2 +z 2 =1. (Just set up an appropriate iterated integral; don't evaluate.) Also find the divergence of E.

Fri., Feb. 16 : Stokes Theorem and Ampere's Law. Read before class : Hand-out (IMPS supplement). Homework due today: Stewart 17.8: 1, 2, 9, 14 (to save yourself work in the surface integral if you use spherical coordinates, do the theta integration first. Think why this is a good plan!). IMPS supplement: 1 (do just the first two rows in Firgure 1), 4 (part 3) (use Green's Theorem), 5 (part 1) and the following problem: (a) Suppose that F is the curl of another vector field, say F = curl(V). Without doing any computation, explain why Stokes' Theorem and the Divergence Theorem together tell you that div(F) = 0. (b) Write V=(P,Q,R) and compute div(curl V) directly, thus verifying the conclusion of part (a).

 Mon., Feb. 19 : Review. Homework due today: IMPS supplment: 1 (last row), 5 (parts 2, 3,4) (Note on part 3: No orientation is specified so just find the circulation up to sign), 6, 7.

 Tues., Feb. 20 : Exam, 3 - 6 P. M. in 102 Bradley. This is a two-hour exam. You may arrive anytime between 3 and 4 and stay for two hours.

Wed Feb. 21 (or x-hour Feb. 22, to be announced) : Final look at conservative vector field. Review of differentiation, gradients and chain rule. Read before class: Stewart 175, Review Stewart 15.4 (up to p. 944), 15.5, 15.6 . Homework due today: None.

Fri., Feb. 23 : Gradient, divergence and curl in other coordinates. Begin optiimization, critical points of functions of more than one variable. Read before class : Hand-out on gradient and divergence, and Stewart 15.7 (skip for now the part beginning with the red box on p. 974, to the last paragraph on p. 978. We will cover that part on Monday.) Homework due today: Stewart Section 17.5: 15, 16, 18, 20, 21. Section 15.5: 4, 8, 41. Section 15.6: 10, 31, 36.

Mon., Feb. 26 : Optimization (continued). Review tangent planes to level surfaces. Read before class : Stewart : remainder of section 15.7. Review p. 967 (middle of page) to 970. Homework due today: Problems 1, 2 on hand-out. Stewart Section 15.7: 3 (Ignore last part about the Second Derivative Test), 6 (ignore instructions; just find the critical points), 14 (just find the critical points), 17 (just find the critical points). Section 17.3: 15.

Wed., Feb. 28 : Wave equation. Review harmonic oscillator equation. Read before class : Stewart: p. 937 and Section 17.5, especially pages 1113-1115. Also review 18.1 and 15.6 from middle of page 967 to end. (We will not discuss 15.6 in class but there will be homework problems on this part for Friday. We will need this material as preparation for Lagrange mulitpliers, to be discussed Friday.) Homework due today: Stewart Section 15.7: 4, 10, 17 (you found the critical points last time but are now to complete the problem),18, 40. Also do the exercise handed out in class on Feb. 26.

Friday March 2 : Stewart 15.8. Homework due today: Stewart Section 15.3: 68 (a, c), 69. Section 15.6: 40(a), 46. Section 17.5: 12, 29 (this problem is tedious, but the identity you prove here is needed for Friday's physics lecture). Section 18.1: 2, 10, 12, 33. Also do the following problem: Consider a vibrating string of length L with its ends held fixed. Under idealized conditions, small vibrations of this string are governed by t he wave equation (on p. 937), where the function u depends on time t and position x on the string; here x ranges over the interval [0,L]. Since the ends of the string are held fixed, any solution u is required to satisfy the boundary conditions that u(t,0) =0=u(t,L) for all t. Using your work on problem 33 in 18.1, find all values of k for which the solution of the wave equation found in 68(a) in 15.3 satisfies these boundary conditions. Each such solution is a standing wave solution. What is its frequency of vibration?

Mon., March 5 : Finish up optimization and any other catch-up work. Then ask: Can you hear the shape of a drum? Read before class. no new reading. Homework due today: Stewart Section 15.8: 1, 2 (If you don't have a suitable graphing calculator, just graph by hand enough level curves to see why the solutions you get using Lagrangian multipliers are what one might expect), 9, 18, 28, 36.

Wed., March 6 : Review. Read before class. no new reading. Homework due today: Stewart 15.7: 29, 33 (you can check the boundary either by using Lagrange multipliers or directly). Handout on "Gradient and Divergence in Polar and Cylindrical Coordinates", problem 3. (Note: The homework will not be collected but you should work out the problems before class.)

PRACTICE PROBLEMS: Stewart chapter 15 review exercises: 50, 54, 63. Chapter 16 review exercises: 12, 31, 38, 41, 47. Chapter 17 review exercises: 14, 23, 26(c), 31, 33. Also t he following problems:

  1. Suppose that the charge density inside the unit ball x2+y2 +z2 <1 is given at each point (x,y,z) by exp(-(x2+y2 +z2)3/2). There is no charge outside the ball. Find the flux of the resulting electric field across the sphere of radius R for each R. Then (using a symmetry argument), find the magnitude and direction of the electric field and express the electric field in spherical coordinates. What is the divergence of this electric field

FRIDAY MARCH 9 : HELP SESSION. Bradley 102.

MONDAY MARCH 12:, 9:00, BRADLEY 102: FINAL EXAM

 

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