MATH 318 : Calculus of Several Variables (Fall 2010):


Instructor: Alvaro Pelayo, 212 Cupples I, apelayo@math.berkeley.edu

Lectures: Monday, Wednesday, Friday 1-2pm, Room 113 Cupples I

Office hours: Monday 2-3:30pm, Friday 2:30-4pm at 212 Cupples I

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 September 29, 2010 (25%), Midterm Exam 2 on November 10 (25%), cumulative Final Exam (35%) on December 22. 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): "Review of Matrices. Continuity of functions of several variables. Partial derivatives, gradient. Maximum value theorem, Lagrange multipliers. Contraction mappings, implicit function theorem. Differential forms, integration theory." - as in (parts of) chapters 1-9 of the book. Optional topics (time permitting) may include spectral theory and geometric aspects of differential equations.

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 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 Sep 1): 1.1 (vectors in R^n) Homework 1 (sec. 1.1-1.4, 2.1, 2.2)

Lecture 2 (Fr Sep 3): 1.2 (Cauchy-Schwarz inequality, triangle inequality)

Week 2:

Monday September 6 is a holiday

Lecture 3 (We Sep 8): 1.3 (subspaces of R^n)

Lecture 4 (Fr Sep 10): 1.4 (linear transformations and matrix algebra)

Week 3:

Lecture 5 (Mo Sep 13): 2.1 (functions of several variables), 2.2 (topology of R^n)

Lecture 6 (We Sep 15): 2.2 (continuation) Deadline to drop a course with no permanent record notation September 15

Lecture 7 (Fr Sep 17): 2.3 (limits and continuity in several variables, epsilon-delta definition)

Week 4:

Lecture 8 (Mo Sep 20): 2.3 (continuation) Homework 1 due, Homework 2 (sec. 2.2, 2.3, 3.1, 3.2)

Lecture 9 (We Sep 22): 2.3 (continuation)

Selected proofs for Midterm 1

Lecture 10 (Fr Sep 24): 3.1 (direccional derivative), 3.2 (differentiablility)

Week 5:

Lecture 11 (Mo Sep 27): 3.2 (continuation)

Lecture 12 (We Sep 29): no lecture, instead Midterm Exam 1 on Lectures 1-10 (Wednesday September 29, 2010) during class time.

Lecture 13 (Fr Oct 1): Correction of Midterm Exam 1 on the blackboard.

Week 6:

Lecture 14 (Mo Oct 4): 3.3 (differentiation rules) Homework 2 due, Homework 3 (sec. 3.3, 3.4)

Lecture 15 (We Oct 6): 3.3 (differentiation rules)

Lecture 16 (Fr Oct 8): 3.4 (the gradient), 3.5 (theory of curves)

Week 7:

Lecture 17 (Mo Oct 11): 3.5 (continuation)

Lecture 18 (We Oct 13): 3.6 (higher order partial derivatives, harmonic functions), catch-up

Friday Oct 15 is a holiday

Week 8:

Lecture 19 (Mo Oct 18): review of 4.1, 4.2 (linear systems of equations, inverse matrices) Homework 3 due, Homework 4 (sec. 3.5, 4.1, 4.2, 4.3, 4.4)

Lecture 20 (We Oct 20): review of 4.3, 4.4 (basis, dimension, subspaces)

Lecture 21 (Fr Oct 22): review of 4.3, 4.4 (continuation)

Week 9:

Lecture 22 (Mo Oct 25): 4.5 (introduction to manifolds)

Lecture 23 (We Oct 27): 5.1 (compactness,convergence theorems)

Lecture 24 (Fr Oct 29): 5.1 (Maximum value theorem, uniform continuity theorem) Homework 4 due, Homework 5 (sec. 4.5, 5.1, 5.2)

Week 10:

Lecture 25 (Mo Nov 1): 5.2 (Maxima, minima, critical points)

Lecture 26 (We Nov 3): 5.3 (Second derivative theorem)

Selected proofs for Midterm 2

Lecture 27 (Fr Nov 5): 5.3 (continuation)

Week 11:

Lecture 28 (Mo Nov 8): 5.4 (Lagrange multipliers) Homework 5 due, Homework 6 (sec. 5.3, 5.4, 5.5)

Lecture 29 (We Nov 10): no lecture, instead Midterm Exam 2 on Lectures 11-26 (Tuesday November 10, 2010) during class time.

Lecture 30 (Fr Nov 12): 5.5 (Gram-Schmidt theorem, Lagrange interpolation theorem)

Week 12:

Lecture 31 (Mo Nov 15): 6.1 (contraction mapping theorem)

Lecture 32 (We Nov 17): 6.2 (inverse function theorem)

Lecture 33 (Fr Nov 19): 6.2 (implicit function theorem) Homework 6 due, Homework 7 (sec. 6.1, 6.2, 6.3)

Week 13:

Lecture 34 (Mo Nov 22): 6.3 (Manifolds in R^n)

Wednesday Nov 24 is a holiday

Friday Nov 26 is a holiday

Week 14:

Lecture 35 (Mo Nov 29): 7.1 (partititons, integral sums, integrability of functions)

Lecture 36 (We Dec 1): 7.1 (continuation) Homework 8 (sec. 7.1, 7.2, 7.3, 7.4)

Lecture 37 (Fr Dec 3): 7.2 (Fubini's theorem) Homework 7 due

Week 15:

Lecture 38 (Mo Dec 6): 7.3 (coordinates), 7.5 (determinants, n-dimensional volume)

Lecture 39 (We Dec 8): 7.6 (change of variables theorem) Homework 9 (sec. 7.5, 7.6, not to be turned in)

Lecture 40 (Fr Dec 10): 8.1, 8.2 (differential forms) Last day of classes. Homework 8 due Friday December 10, before 5pm, at 212 Cupples I. You can also turn it in at Friday's class.

Final Exam December 22.