MATH 0350: HONORS
CALCULUS (Section 2, 13519)
Professor Steven J. Miller (sjmiller AT math.brown.edu), Kassar House, Room 210, 401-863-1123
MIDTERM EXAM: TBD
FINAL
EXAM: Wednesday, December 5th, 90 mins
from 7 to 9am plus take-home component.
COURSE DESCRIPTION: A third-semester calculus course for students of greater aptitude and motivation. Topics include vector analysis, multiple integration, partial differentiation, line integrals, Green's theorem, Stokes' theorem, the divergence theorem, and additional material selected by the instructor. Prerequisite: Advanced placement or written permission. NOTE: This is an honors course. We will cover the material in greater depth than Math 0180 and 0200, and will be moving at a very fast pace. You should expect to spend at least one if not two hours a day (every day!) on this course. I strongly encourage you to work in groups, and you should skim the reading before each class. We will not cover all the material in the book in class; you are responsible for reading the examples at home. While we will do some examples in class, we will often concentrate on the proofs, theory, and some of the more interesting examples in class.
The textbook is the fifth edition of Vector Calculus by Jerrold E. Marsden and Anthony Tromba (ISBN: 0-7167-4992-0, ISBN-13: 978-0-716-74992-9), as well as supplemental handouts. Please read the relevant sections before class. This means you should be familiar with the definitions as well as what we are going to study; this does not mean you should be able to give the lecture. We will cover every section in order, except for 7.7 and 8.5. You do not need a calculator for this class, though I strongly urge you to become familiar with either Matlab or Mathematica to plot some of the multi-dimensional objects. There are many good references on the web. In addition to the ebrary site I mentioned, you can access certain books online: Calculus in Vector Spaces (Lawrence J. Corwin, Robert Henry Szczarba) and Multivariable Calculus (Lawrence J. Corwin, Robert Henry Szczarba).
GRADING / HW: Homework 20%, Midterms 40% (best 3 of 5 exams), Final 40%. Exams are black tie optional. Homework is to be handed in on time, stapled and legible. Late, messy or unstapled homework will not be graded. I encourage you to work in small groups, but everyone must submit their own homework assignment. Extra credit problems should not be included in the general homework, but handed in separately. Very little partial credit is given on these problems.
OFFICE HOURS: Primary office hours: MWF 8 to 9am; Secondary office hours: before and after class (though I have more time before). If possible email me ahead of time to let me know you are coming, as well as what you want to discuss.
HANDOUTS:
Handout on Types of Proofs (from a handout I wrote for math review sessions at Princeton, 1996-1997; this was written for students from calculus to linear algebra).
Intermediate and Mean Value Theorems and Taylor Series (you should know this material already; the main results are stated and mostly proved, subject to some technical results from analysis which we need to rigorously prove the IVT).
The chain rule: this is a slight modification of a handout I wrote for a similar class at Princeton in the mid-90s. Unfortunately it is in MSWord.
Equality of mixed derivatives: two articles on the web: here and here (the second link is through JStor, and might only work when you are at Brown).
The arithmetic and geometric mean: a handout I wrote a few years ago for a freshmen semianr at Brown.
my paper A probabilistic proof of Wallis' formula for π, to appear in the American Mathematical Monthly (there are a lot of good articles in this magazine, many of which are accessible to freshmen).
Please spend at least 1 if not 2 hours a night
reading the material/looking at the proofs/making sure you can do the algebra.
Below is a tentative reading list and homework assignments. It is subject to
slight changes depending on the amount of material covered each week. I strongly
encourage you to skim the reading before class, so you are familiar with the
definitions, concepts, and the statements of the material we'll cover that day.
Week One (9/5 - 9/7):
Read: Chapter 1.
HW: Due 9/12 (though I strongly urge you to have it done by 9/10 if
possible): Page 22: #14, #17, #26, prove that the product of the slopes of
two perpendicular lines is -1 (so 0 * oo = -1 if you consider the x- and y-axes);
Page 36: #17, #18, prove that the sum of the squares of the lengths of two diagonals of a parallelogram
equals the sum of the squares of the lengths of the four sides (so if the
sides have lengths a,b,c,d and the diagonals have lengths e, f then a^{2}
+ b^{2}
+ c^{2}
+ d^{2}
= e^{2}
+ f^{2}.). Page 61: #2c, #6, #21a; Page 72: #4.
Suggested Problems:
Page 22: #28; Page 36: #3, #14, #20; #16, #21bc, #22, #25, prove the distance
formula (ie, the generalization of the Pythagorean formula) in n-dimensions,
prove that the n x n matrix with first row 1, 2, ..., n, ..., and last row n^{2}
+ 1, ..., n^{2} has determinant 0 if n > 2; Page 90: #19.
Extra Credit: Due 9/19: You do not need to do all of them; however, very
little partial credit will be awarded (extra credit is basically right or
wrong).
(1) Let N be a large integer. How should we divide N into positive integers a_{i} such that the product of the a_{i} is as large as possible. Redo the problem when N and the a_{i} need not be integers.
(2) What is wrong with the following argument (from Mathematical Fallacies, Flaws, and Flimflam - by Edward Barbeau): There is no point on the parabola 16y = x^{2} closest to (0,5). This is because the distance-squared from (0,5) to a point (x,y) on the parabola is x^{2} + (y-5)^{2}. As 16y = x^{2} the distance-squared is f(y) = 16y + (y-5)^{2}. As f'(y) = 2y+6, there is only one critical point, at y = -3; however, there is no x such that (x,-3) is on the parabola. Thus there is no shortest distance!
(3) Without using any
computer, calculator or computing by brute force, determine which is larger: e^{π}
or π^{e}. (In other words, find out which is larger without
actually determining the values of
e^{π} or π^{e}).
Week Two (9/10 - 9/14):
Read: Chapter 2; Section 2.1 and
2.2 for Monday, 2.3 and 2.4 for Wednesday and the rest for Friday. Read
the chain rule handout.
HW: Due 9/17: Page 105: #1, #24. Page 125: #4, #8, #17, #21. Page 139:
#2a, #4a, #5, #7c, #12a, #13a. Page 149: #2, #6, #15.
Page 159: #2g, #7, #25(iv).
Suggested Problems: Page
105: #3, #30. Page 125: #9, #10, #18, #23, #25, #26, #27. Page 139: #3, #4cde,
#10, #15, #18, #19.
Page 149: #3, #7, #13, #17. Page 159: #1, #4.
Week Three (9/17 - 9/21):
Read: For
Monday: Sections 2.5, 2.6, 3.1; for Wednesday: Sections 3.1, 3.2, 3.3; for
Friday: Sections 3.3, 3.4, 3.5 (just read the statements).
HW: Due 9/24: Page 159:
#12, #20, #24.
Page 171: #2ab, #4a, #6a, #16. Page 176: #23, #47. Page 191: #1, #11. Page
202: #2, #3, #7a.
Suggested Problems:
Page 159: #8, #17, #18, #26, #28.
Page 171: #5a, #12, #17, #21, #23. Page 176: #26, #41, #42. Page 191: #16,
#19. Page 202: #7bcd.
Week Four (9/24 - 9/28): in-class exam on Friday
(closed book, no calculators)
Read: Sections
3.3, 3.4, 3.5 (just read the statements in 3.5), 4.1, 4.2.
HW:
Due 10/1: Page 222: #7, #22, #36.
Page 243: #2, #10.
Suggested Problems:
Page 222: #18, #23, #28, #41. Page 243: #20, #27.
Week Five (10/1 - 10/5):
Read: 4.2, 4.3, 4.4, 5.1, 5.2.
HW: Due 10/10: Page 273: #1, #7, #13. Page 281: #1, #11, #13. Page 293: #5. Page 310: #3, #15, #31. Page 325: #1ac, #6.
Suggested Problems: Page 273: #14, #15, #20, #21. Page 281: #3, #12, #14, #15, #16, #17, #19. Page 293: #2, #7, #20.
Page 310: #25. Page 325: #1bd, #3.
Week Six (10/8 - 10/12):
no class on Monday b/c of Columbus day, second exam Mon 10/15 for 90 mins
b/w 7 and 9am in Kassar 105
Read: 5.2, 5.3, 5.4, 5.5.
HW: Due 10/15: Page 339: #1ab, #8, #9, #12. Page 347: #2bd, #3, #15.
Suggested Problems:
Page 339: #1cd, #11. Page 347: #9, #12, #16.
Week Seven (10/15 - 10/19):
second exam Mon 10/15 for 90 mins b/w 7 and 9am in Kassar 105
Read: Xerox
notes on Monte Carlo Integration, 6.1, 6.2, 6.3.
HW: Due 10/22: Page
353: #1b, #4, #14. Page 363: #3, #9, #12. Page 375: #2, #7, #12. Page 390: #2,
#3a, #11, #29, #34.
Suggested Problems: Page 353: #6, #15. Page
363: #8, #19, #27. Monte Carlo Exercises: (1) write a program to estimate the
area of a quarter of the unit
circle; (2) let f be a probability distribution with finite mean and variance.
Prove there exist a and b such that (i) g(x) = f(x+a) has mean zero and the
same standard deviation as f, and (ii)
h(x) = g(bx) = f(b(x+a)) has mean 0 and variance 1. This result is important
because it normalizes different distributions and makes them more easily
comparable. For a nice distribution, you don't really start to see the shape
until the third or fourth moment (in general). This feature is responsible for
some incredibly universality of behavior (see the Central Limit Theorem in
Math 162). Page 375: #3, #8. Page 390: #4, #7, #10, #15, #18, #31.
Extra Credit: Due 11/5: (1: 1 point) Prove Newton's result that you may
assume all the mass of a sphere of radius 1 with uniform density is
concentrated at the center. As the book proves this using potentials,
you must prove this by direct integration of the force. (2: 2 points) Assume the force of gravity is given
by (GMm/r^{n-1}) e_{r }in n-dimensional space.
Here e_{r} is the unit vector in the r-direction. Thus the
magnitude of the force is GMm/r^{n-1} and it is radial. Prove or disprove: we
may assume all the mass of a sphere of radius 1 with uniform density is
concentrated at the center (ie, the force this exerts is the same as the force
of the sphere).
Week Eight (10/22 - 10/26): Exam 3: Wednesday, 10/24
for 90 mins b/w 7 and 9am in Kassar 105. The exam is cumulative, but there
will be no problem on Lagrange Multipliers.
Read: 6.2,
6.3, 6.4, 7.1. If you're interested in formulas for
π, see also my paper
A probabilistic proof of Wallis'
formula for π,
to appear in the
American Mathematical Monthly (there are a lot of good articles in this
magazine, many of which are accessible to freshmen).
HW: Due 10/29: Page 404: #1. Page 415: #5, #17, #18.
Suggested Problems:
Page 404: #3, #17, #18. Page 415: #1, #12, #13.
Week Nine (10/29 - 11/2):
Read: 7.1,
7.2, 7.3, 7.4.
HW: Due 11/5:
Page 427: #3a, #6, #13. Page 447: #1ad, #4, #11, #14. Page 459: #1, #9a, #16.
Page 480: #1, #6, #11ab, #17 (Note: for #17, recall that a harmonic function
is defined on page 192, #19, and the upsidedown triangle squared is defined on
page 305: Ñ^{2}
Phi =
Ñ · (Ñ Phi), and is
called the Laplacian of Phi).
Suggested Problems:
Page 427: #4a, #8, #12, #16. Page 447: #3, #5, #13, #17. Page 459: #4, #10,
#13, #14, #17, #18. Page 471: #1, #2, #16, #23.
Week Ten (11/5 - 11/9): Next exam will be
Friday, November 16th (90 mins b/w 5 and 7pm in Kassar 105).
Read: 7.5,
7.6, 8.1, 8.2, 8.3.
HW: Due 11/12: Page 497: #3, #8, #11. Page 528: #1, #5,
#11, #12, #16. Page 547: #1, #14 (do not do the physical interpretation), #17,
#23.
Suggested Problems:
Page 480: #5, #7, #15, #16. Page 497: #5, #14, #15. Page 528: #2, #6, #8, #15,
#18, #19, #21 thru #27.
Page 547: #5, ,#6, #7, #8, #15, #18, #20, #22.
Week Eleven (11/12 - 11/16): Exam will be Friday,
November 16th (90 mins b/w 5 and 7pm in Kassar 105).
Read: 8.3,
8.4, 8.5
HW: Due 11/19: Page 558: #1, #2, #3, #12 (important), #13ac, #16. Page
573: #1, #7, #12, #15, #18, #19, #23.
Suggested Problems:
Page 558: #4, #5, #6, #11, #15, #17,
#25. Page 573: #2, #5, #11, #14, #16, #17, #22.
Week Twelve (11/19): No class on Wednesday or Friday
Read:
HW: Due 11/26: think about mathematics.
Week Thirteen (11/26 - 11/30): Exam will be Wednesday,
November 28th from 7 to 9am in Kassar 105 (any 90 minute block).
Read: 8.6.
HW: no written homework; study for exams. It is strongly recommended
that you do practice problems (either suggested problems from various
assignments throughout the course, or other problems in the textbooks
available through ebrary).
Week Fourteen (12/3 - 12/7): Friday, December 7th is the
last day of classes. There will be an exam on Wednesday, December 5th,
7 to 9am in Kassar 105 (any 90 minute block).
Read:
HW:
no written homework; study for exams. It is strongly recommended
that you do practice problems (either suggested problems from various
assignments throughout the course, or other problems in the textbooks
available through ebrary).
Please spend at least 1 if not 2 hours a night reading the material/looking at the proofs/making sure you can do the algebra. Below is a tentative reading list and homework assignments. It is subject to slight changes depending on the amount of material covered each week. I strongly encourage you to skim the reading before class, so you are familiar with the definitions, concepts, and the statements of the material we'll cover that day.