MATH 318 : Calculus of Several Variables (Fall 2012)


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

Lectures: Monday, Wednesday, Friday 1-2pm, Room DUNCKER 101

Office hours: Monday 4:30-5:30pm, Wednesday 10-11:30am

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 Wednesday October 3 (25%), Midterm Exam 2 on Wednesday November 7 (25%), cumulative Final Exam (35%) on December 19. 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 (Wed Aug 29): 1.1 (vectors and matrices) Homework 1 (sec. 1.1, 1.2, 1.3)

Lecture 2 (Fr Aug 31): 1.1 (continuation)

Week 2:

Monday September 3 is a holiday

Lecture 3 (Mo Sep 5): 1.2 (Cauchy-Schwarz inequality, triangle inequality)

Lecture 4 (We Sep 7): 1.3 (subspaces of R^n)

Week 3:

Lecture 5 (Mo Sep 10): 1.4 (linear transformations and matrix algebra)

Lecture 6 (We Sep 12): 1.4 (continuation) Deadline to drop a course with no permanent record notation September 12 Homework 1 due Homework 2 (sec. 1.4, 2.1)

Lecture 7 (Fr Sep 14): ) 2.1 (functions of several variables)

Week 4:

Lecture 8 (Mo Sep 17): , 2.2 (topology of R^n)

Lecture 9 (We Sep 19): 2.2 (continuation)

Selected proofs for Midterm 1

Lecture 10 (Fr Sep 21): 2.3 (limits and continuity in several variables, epsilon-delta definition) Homework 2 due, Homework 3 (sec. 2.2, 2.3)

Week 5:

Lecture 11 (Mo Sep 24): 2.3 (continuation)

Lecture 12 (We Sep 26): 2.3 (continuation)

Lecture 13 (Fr Sep 28): 3.1 (direccional derivative)


Week 6:

Lecture 14 (Mo Oct 1): 3.2 (differentiablility)

Lecture 15 (We Oct 3): no lecture, instead Midterm Exam 1 on Lectures 1-12 (Wednesday October 3, 2012) during class time.

Lecture 16 (Fr Oct 5): Correction of Midterm Exam 1 on the blackboard

Week 7:

Lecture 17 (Mo Oct 8): 3.2 (continuation) Homework 3 due, Homework 4 (sec. 3.1, 3.2, 3.3)

Lecture 18 (We Oct 10): 3.2 (continuation)

Lecture 19 (Mo Oct 12): catch up

Week 8:

Lecture 20 (We Oct 15) :3.3 (Differentiation rules)

Lecture 21 (Fr Oct 17): 3.3. (continuation)

Friday Oct 19 is a holiday

Week 9:

Lecture 22 (Mo Oct 22): 3.4 (the gradient), 3.5 (theory of curves)

Lecture 23 (We Oct 24): 3.5 (continuation) Homework 4 due, Homework 5 (sec. 3.3, 3.5, 3.6)

Lecture 24 (Fr Oct 26): 3.6 (higher order partial derivatives, harmonic functions)

Week 10:

Lecture 25 (Mo Oct 29): review of 4.1, 4.2 (linear systems of equations, inverse matrices)

Lecture 26 (We Oct 31): review of 4.3, 4.4 (basis, dimension, subspaces)

Selected proofs for Midterm 2

Lecture 27 (Fr Nov 2): 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)


Week 11
:

Lecture 28 (Mo Nov 5): 5.1 (compactness,convergence theorems)

Lecture 29 (We Nov 7): no lecture, instead Midterm Exam 2 on Lectures 13-26 (Wednesday November 7, 2012) during class time.

Lecture 30 (Fr Nov 9): Correction of Midterm Exam 2 on the blackboard.

Week 12:

Lecture 31 (Mo Nov 12): 5.1 (Maximum value theorem, uniform continuity theorem)

Lecture 32 (We Nov 14): 5.1 (continuation)

Lecture 33 (Fr Nov 16): catch up and/or in class practice problems concerning 5.1 Homework 6 due, Homework 7 (sec. 4.5, 5.1, 5.2)

Week 13:

Lecture 34 (Mo Nov 19): catch up and/or in class practice problems concerning 5.1

Wednesday Nov 21 is a holiday

Friday Nov 23 is a holiday

Week 14:

Lecture 35 (Mo Nov 26): 5.2 (Maxima, minima, critical points)

Lecture 36 (We Nov 28): 5.2 (continuation) Homework 8 (sec. 5.3)

Lecture 37 (Fr Nov 30): 5.3 (Second derivative theorem) Homework 7 due

Week 15:

Lecture 38 (Mo Dec 3): 5.3 (continuation)

Lecture 39 (We Dec 5): 6.1 (contraction mapping theorem)

Lecture 40 (Fr Dec 7): 6.1 (continuation and practice problems) Homework 8 due. Last day of classes. Homework 8 due before 5pm, at 212 Cupples I. You can also turn it in at Friday's class.

Final Exam December 19, 2012.