Math 222 Course Webpage

Vector Calculus

Spring 2015, Duke University


General information | Course description | Assignments | Homework schedule | Grading | Links | Fine print

General information

Goal: Students will become proficient in multivariable calculus, including differential, integrable, and vector calculus.

Lectures: Tuesday and Thursday

Text: Vector Calculus, by Jerrold Marsden and Anthony Tromba, sixth edition.

Course content: Chapters 1 – 8 of the course text, by Marsden & Tromba

Prerequisite: Math 221 (Linear Algebra) and its prerequisites (second-semester Calculus: Math 122, 112L, or 122L)

Contact information for the Instructor

Name: Professor Ezra Miller
Address: Mathematics Department, Duke University, Box 90320, Durham, NC 27708-0320
Office: Physics 209
Phone: (919) 660-2846
Email: ezramath.duke.edu
Webpage: http://math.duke.edu/~ezra
Course webpage: you're already looking at it... but it's http://math.duke.edu/~ezra/222/222.html
Office hours: Monday 15:00 – 16:00 & Tuesday 16:20 – 17:15, in Physics 209


Assignments

Policies regarding graded work


Homework schedule

If a lecture or assignment hasn't been posted, and you think it should have been, then please do email me. Sometimes I encounter problems (such as, for example, the department's servers going down) while posting assignments; other times, I might simply have forgotten to copy the updated files into the appropriate directory, or to set the permissions properly.

Read and study the text carefully before attempting the assignments. Make sure you fully understand the given proofs and examples; note that there are examples in the text similar to most of the homework problems. The material in gray shaded boxes consists of definitions and theorems; learn them precisely. (Often when students say they do not know how to do a problem it is because they don't know the definitions of the terms in the problem.) If you have trouble understanding something in the text after working on it for a while, then see me in office hours or e-mail me.

# Date Sections Topic Assignment #: Due date
  1. Thu  8 Jan 1.1–1.3, 1.5
2.1
review of linear algebra
introduction to level sets
p.18: 17;  p.29–30: 7, 8, 30
 
#1: 20 Jan
  2. Tue 13 Jan 2.1
2.2
visualizing functions: graphs and sections
open sets
p.86–87: 7, 17, 18, 21
 
  3. Thu 15 Jan 2.2 limits and continuity p.103–104: 12 (justify your answers), 24, 26 (use ε-δ), 27, 33, 34
  4. Tue 20 Jan 2.3 differentiation as a linear map p.115–116: 4ab, 5, 10, 16, 22, 28 #2: 27 Jan
  5. Thu 22 Jan 2.4
4.1
paths, curves, and velocity vectors
acceleration and Newton's second law
p.123–124: 7, 8, 15, 19, 20
p.227–228: 1, 4
  6. Tue 27 Jan 4.1
4.2
Newton's law of gravitation, circular orbits, Kepler's law
arclength
p.227–228: 9, 14, 18, 20, 24
p.234–235: 2, 3, 16 (use rules on p.218), 17, 20
#3: 3 Feb
  7. Thu 29 Jan 2.5 product, sum, chain rules for differentiation
 
p.132–133: 3bd, 6, 8, 17, 19 (give details!), 35
(optional challenge problem: p.134: 28)
  8. Tue  3 Feb 2.6, 4.3
review
gradients, directional derivatives, tangent planes to level surfaces
Review problems for Exam 1
p.142–143: 17, 22, 24, 26
p.144–145: 14, 25, 27, 31
#4: 10 Feb
  9. Thu  5 Feb 3.1
3.2
iterated partials, equality of mixed partials, heat and wave equations
Taylor's theorem, estimate on remainder
p.156–158: 4, 9, 10, 22, 31
p.166: 7, 8; Use #8 to approximate f(x,y) at x = 1.06 and y = –.02
10. Tue 10 Feb 3.3 extrema, critical points, Hessians, positive definiteness p.182–184: 1, 21, 23, 27, 30, 40, 43, 46 #5: 17 Feb
  Thu 12 Feb Midterm Exam 1 (covers all material treated through 5 February)
  Tue 17 Feb (snow day)
11. Thu 19 Feb 3.4 constrained extrema, Lagrange multipliers, bounded closed sets p.201–202: 5, 6, 13, 18, 20 (use Lagrange multipliers for these), 22, 31, 32 #6: 3 Mar
12. Tue 24 Feb 3.4 Lagrange mult: multiple constraints; global max/min on bounded regions
  Thu 26 Feb (snow day)
13. Tue  3 Mar 3.5 implicit & inverse function thms, derivatives of implicit functions p.210–211: 2, 3, 8, 12, 13, 16, 19 (warm up: try the case n=2 first) #7: 17 Mar
14. Thu  5 Mar 5.1
5.2
integrals over rectangles; Cavalieri's principle
iterated integrals; Fubini's theorem
p.270–271: 3, 7, 10
p.282: 1, 5, 9, 12 (think—but don't write—about the last sentence), 15
  Tue 10 Mar no class: Spring break
  Thu 12 Mar
15. Tue 17 Mar 5.3–5.4
5.5
integrals over more general regions; changing order of integration
triple integrals
p.288–289: 3, 16, 20;  p.293: 3abc, 4, 14, 17
p.303–304: 12, 24, 27
#8: 24 Mar
16. Thu 19 Mar 6.1
6.2, 1.4
one-to-one and onto mappings, polar & spherical coordinates
change of variables formula
p.313: 7 (also draw a picture of D), 8
p.327–328: 3, 5, 10, 25, 26, 35
17. Tue 24 Mar 6.3
4.3
applications: average, center of mass, moment of inertia
vector fields, flow lines, gradient vector fields
p.337–338: 4, 7, 14, 17, 21
p.243–244: 2, 5, 9, 10, 16, 26
#9: 31 Mar
  Thu 26 Mar Midterm Exam 2 (cumulative, but emphasizing material between Exam 1 and 19 March)
18. Tue 31 Mar 4.4 cross product, divergence, curl, and physical interpretations
p.258–260: 1, 2, 5, 13, 15, 26, 37 #10: 7 Apr
19. Thu  2 Apr 7.1
7.2
integrals over paths, reparametrization
work done by a force field on a particle moving along a path
p.356–358: 10, 13, 17, 20, 26 29 (typo: the correct definition of T is ∫C(1/v)ds
where C is the given path from A=(0,1) to B=(1,0));  p. 373: 5
20. Tue  7 Apr 7.2 line integrals (i.e., integrating vector fields over curves) p.373–5: 4, 6, 11, 13 (justify), 16, 17, 19 #11: 14 Apr
21. Thu  9 Apr 7.3
7.4
parametrized surfaces, tangent planes
surface area
p.381–383: 7, 11, 14, 15, 19, 20, optional challenge problem: 24
p.391–392: 3, 12, 25
22. Tue 14 Apr 7.5
7.6
integrals over surfaces, independence of parametrization
flux, oriented surfaces
p.398: 4, 7, 9, 14
p.412–413: 8, 10, 11 (redefine the last component of F to (z-x2)k), 20, 22     
#12: 21 Apr
23. Thu 16 Apr 8.1
8.2
oriented boundaries of planar domains, Green's Theorem
Stokes' Theorem
p.437–438: 7, 9, 11ab, 20, 23
p.451–452: 13 (assume S1 and S2 oriented with n outward pointing), 14, 23
24. Tue 21 Apr 8.3
8.4
conservative vector fields, which vector fields are gradient fields?
divergence theorem and applications
suggested exercises (won't be graded): p.460–461: 13, 16, 17, 27, 28
suggested exercises (won't be graded): p.474–475: 7, 9, 11, 24, 28
no due date
  Thu 23 Apr review session, 15:00 – 17:00
  Mon 27 Apr 222-02 final exam, 14:00 – 17:00
  Tue 28 Apr 222-01 final exam, 19:00 – 22:00


Grading scheme

Final course grades: Quizzes and participation in class discussion can contribute to your homework score.


Links

University academic links Departmental links


The fine print

I will do my best to keep this web page for Math 222 current, but this web page is not intended to be a substitute for attendance. Students are held responsible for all announcements and all course content delivered in class.
Many thanks are due to Jeremy Martin and Vic Reiner, who provided templates for this webpage.


The views and opinions expressed in this page are strictly those of the page author. The contents of this page have not been reviewed or approved by Duke University.

ezramath.duke.edu
Wed Apr 15 21:01:04 EDT 2015


Back to homepage    Math Home Page