MATH 318 : Calculus of Several Variables (Winter 2014)


Instructor: Alvaro Pelayo, apelayo@math.wustl.edu

Required Text: Theodore Shifrin, "Multivariable Mathematics. Linear Algebra, Multivariable Calculus, and Manifolds", John Wiley & Sons, 2005.

Other texts:

1. James R. Munkres, "Analysis on Manifolds", Addison-Wesley, 1991.

2. Michael Spivak, "Calculus on Manifolds: A Modern Approach to the Classical Theorems of Advanced Calculus", W.A. Benjamin, 1965.

Grading: homework (15%), Midterm Exam 1 on Monday February 17, 2014 (25%), Midterm Exam 2 on Wednesday March 26, 2014 (25%), cumulative Final Exam (35%) on May 5, 2014. Exams will consist of a few theory questions, including definitions and proofs of selected results, and some problems involving computations and proofs. There will be no make-up exams - if you miss one midterm, the final exam counts 60%. If you take both midterms and your grade in the final is greater than your lowest midterm grade, then the grade in the midterm gets replaced by the grade in the final. For example, if midterm 1 score is 80/100, midterm 2 score is 50/100 and final score is 70/100, your score in the second midterm gets replaced by 70/100. If you choose to be graded "Pass/Fail", a "Pass" grade reqires a grade of C- or higher.

Homework: Assignments can be downloaded from this website, no paper copy will be given in class. A grader will grade selected problems. Discussing homework with others is ok. It is expected that everyone writes in his/her own words the homework solutions. No late homework is accepted and the two lowest homework grades will be dropped (which can also count for missing assignments). Homework is due at the beginning of class on the due date.

Prerequisites: Math 233 and Math 309 (not concurrent), or equivalent knowledge of matrix algebra and multivariable calculus.

Syllabus (approximate): "Vectors and matrices. Continuity and differentiability of functions of several variables. Partial derivatives, gradient. Maximum value theorem. Contraction mappings." - as in (parts of) chapters 1-6 of the book. Optional topics (time permitting) may include implicit function theorem, differential forms or integration.

Tentaive outline: there will be no time to cover all of the sections in each chapter of Shifrin's book and not all sections will be covered in the same depth. Class time is for the fundamental concepts of each chapter. Shifrin's book (and also the other books) are sources for further explanations and examples. The following is the approximate plan *subject to changes* (numbers refer to sections in Shifrin's book); this plan will be updated as the course progresses.


WEEK 1:

Lecture 1 (Mo January 13): 1.1 (vectors and matrices) Homework 1 (sec. 1.1, 1.2, 1.3)

Lecture 2 (We January 15): 1.1 (continuation)

Lecture 3 (Fr January 17): 1.2 (Cauchy-Schwarz inequality, triangle inequality)




WEEK 2
:

Monday January 20 is a holiday

Lecture 4 (We January 22): 1.3 (subspaces of R^n)

Lecture 5 (Fr January 24): 1.4 (linear transformations and matrix algebra)




WEEK 3
:

Lecture 6 (Mo January 27): 1.4 (continuation) Homework 1 due Homework 2 (sec. 1.4, 2.1)

Lecture 7 (We January 29): 2.1 (functions of several variables) Deadline to drop a course with no permanent record notation

Lecture 8 (Fr January 31): , 2.2 (topology of R^n)




WEEK 4
:

Lecture 9 (Mo February 3): 2.2 (continuation)

Selected proofs for Midterm 1

Lecture 10 (We February 5): 2.3 (limits and continuity in several variables, epsilon-delta definition) Homework 2 due, Homework 3 (sec. 2.2, 2.3)

Lecture 11 (Fr February 7): 2.3 (continuation)



WEEK 5:

Lecture 12 (Mo February 10): 2.3 (continuation)

Lecture 13 (We February 12): 3.1 (direccional derivative)

Lecture 14 (Fr February 14): 3.2 (differentiablility)




WEEK 6:

Lecture 15 (Mo February 17): no lecture, instead Midterm Exam 1 on Lectures 1-12 (Monday February 17, 2014) during class time.

Lecture 16 (We February 19): Correction of Midterm Exam 1 on the blackboard

Lecture 17 (Fr February 21): 3.2 (continuation) Homework 3 due, Homework 4 (sec. 3.1, 3.2, 3.3)




WEEK 7
:

Lecture 18 (Mo February 24): 3.2 (continuation)

Lecture 19 (We February 26): catch up

Lecture 20 (Fr February 28) :3.3 (Differentiation rules)


WEEK 8:

Lecture 21 (Mo March 3): 3.3. (continuation)

Lecture 22 (We March 5): 3.4 (the gradient), 3.5 (theory of curves)

Lecture 23 (Fr March 7): 3.5 (continuation) Homework 4 due, Homework 5 (sec. 3.3, 3.5, 3.6)


WEEK 9
:

No classes - Spring Break


WEEK 10:

Lecture 24 (Mo March 17): 3.6 (higher order partial derivatives, harmonic functions)

Lecture 25 (We March 19): review of 4.1, 4.2 (linear systems of equations, inverse matrices)

Lecture 26 (Fr March 21): review of 4.3, 4.4 (basis, dimension, subspaces)

Selected proofs for Midterm 2




WEEK 11
:

Lecture 27 (Mo March 24): review of 4.3, 4.4 (continuation), 4.5 (introduction to manifolds) Homework 5 due, Homework 6 (sec. 4.1, 4.2, 4.3, 4.4)

Lecture 28 (We March 26): no lecture, instead Midterm Exam 2 on Lectures 13-25 (Wednesday March 26, 2014) during class time.

Lecture 29 (Fr March 28): Correction of Midterm Exam 2 on the blackboard.



WEEK 12:

Lecture 30 (Mo March 31): 5.1 (compactness,convergence theorems)

Lecture 31 (We April 2): 5.1 (Maximum value theorem, uniform continuity theorem)

Lecture 32 (Fr April 4): 5.1 (continuation)



WEEK 13:

Lecture 33 (Mo April 7): catch up and/or in class practice problems concerning 5.1 Homework 6 due, Homework 7 (sec. 5.1, 5.2)

Lecture 34 (We April 9): catch up and/or in class practice problems concerning 5.1

Lecture 35 (Fr April 11): 5.2 (Maxima, minima, critical points)




WEEK 14
:

Lecture 36 (Mo April 14): 5.2 (continuation) Homework 8 (sec. 5.1)

Lecture 37 (We April 16): 5.3 (Second derivative theorem) Homework 7 due

Lecture 38 (Fr April 18): 5.3 (continuation)




WEEK 15
:

Lecture 39 (Mo April 21): 6.1 (contraction mapping theorem)

Lecture 40 (We April 23): 6.1 (continuation and practice problems)

Lecture 41 (Fr April 25): 6.1 (continuation and practice problems) Homework 8 due.