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 (secondsemester Calculus: Math 122, 112L, or 122L)
Name: Professor Ezra Miller
Address: Mathematics Department,
Duke University, Box 90320,
Durham, NC 277080320
Office: Physics 209
Phone: (919) 6602846
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
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 email 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 εδ), 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 
onetoone 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 (zx^{2})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 S_{1} and S_{2} 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  22202  final exam, 14:00 – 17:00  
Tue 28 Apr  22201  final exam, 19:00 – 22:00 
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