Math 222 Course Webpage

Vector Calculus

Spring 2015, Duke University

General information

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

Lectures: Tuesday and Thursday

11:45 – 13:00, Physics Building 235 (222-01)
15:05 – 16:20, Physics Building 235 (222-02)

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

Weekly homework will be collected and graded.
There will be two in-class exams:
- Thursday 12 February
- Thursday 26 March
There will be a final exam (note reversal of days vs. section numbers):
- Section 222-01: Tuesday 28 April 19:00–22:00 in Physics 235
- Section 222-02: Monday 27 April 14:00–17:00 in Physics 235
Quizzes may be given if homework performance lags.

Policies regarding graded work

Tentative due dates for the homework assignments this semester are listed in the table below, updated as the semester progresses.
You should work on the homework for a section immediately after the class in which it is covered.
Homework for the sections covered in a week will be due and collected at the beginning of the following Tuesday's class.
Late homework will not be accepted.
Missed quizzes or exams: in general there will be no make-ups, but some accommodation may be possible in one of the following four situations: personal emergencies or tragedies, an incapacitating illness, a religious holiday, or varsity athletic participation. Please visit these web-pages now to familiarize yourself with the procedures.
Collaboration on homework is encouraged while you discuss the search for solutions, but when it comes time to write them down, the work you turn in must be yours alone: you are not allowed to consult anyone else's written solution, and you are not allowed to share your written solutions. (It is very easy to tell when solutions have been copied or written together.) If you collaborate, you must indicate—on the homework page—who your collaborators were.
Collaboration of any sort on quizzes, in-class exams, and the final exam is not permitted: you must work completely independently without giving or receiving help from others.
Students are expected to adhere to the Duke Community Standard. You must reaffirm your committment to these standards on all work.
If a student is responsible for academic dishonesty on a graded item in this course, then the student will have an opportunity to admit the infraction and, if approved by the Office of Student Conduct, resolve it directly through a faculty-student resolution agreement; the terms of that agreement would then dictate the grading response to the assignment at issue. If the student is found responsible through the Office of Student Conduct and the infraction is not resolved by a faculty-student resolution agreement, then the student will receive a score of zero for that assignment, and the instructor reserves the right to further reduce the final grade for the course by one or more letter grades—possibly to a failing grade—at the discretion of the instructor.
Computer policy: You may use a computational aid for the homework but I recommend avoiding it as much as possible, particularly since electronic devices of all sorts are not allowed on quizzes, in-class exams, and the final exam. Portable electronic devices that are visible or audible during exams will be confiscated until the exam period ends.
All submitted work (including quizzes and exams, in addition to homework) must be written neatly and legibly. Instead of erasing please use a single line crossout.
If a question requires more than a single expression or equation, your response must be phrased in complete sentences.
The logic of a proof must be completely clear for full credit.
Staple multiple-page submissions or they will be returned ungraded. Paper clips do not suffice; pages can become separated.

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

#	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`-`x`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 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:

15% Homework
25% Midterm #1
25% Midterm #2
35% Final exam

Quizzes and participation in class discussion can contribute to your homework score.

Links

University academic links

list of all courses given in the Math Department in Spring 2015.
Official University calendar for academic year 2014–2015.
Fundamental University policies concerning academic dishonesty and related matters.

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

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