Instructor: David A. Meyer
Office hours: AP&M 7256, Tu 12:00nn-2:00pm, or by appointment
Lecture: Center Hall, room 203, MWF 9:00am-9:50am
Email: dmeyer "at" math "dot" ucsd "dot" edu
Teaching assistant: Dan Schultheis
Office hours: AP&M 5132, M 1:00pm-2:00pm, Th 2:00pm-4:00pm, or by appointment
Section: Center Hall, room 201, Tu 10:00am-10:50am
Email: dschulth "at" math "dot" ucsd "dot" edu
This course is the second in the Mathematics Department's Honors sequence. It covers multivariable calculus: limits and continuity, derivatives in several variables, manifolds, Taylor polynomials, quadratic forms, and integration in several variables.
The three courses in the Honors sequence, Math 31ABC, cover essentially the same material as do Math 20F, 20C, and 20E, respectively, but at a more sophisticated conceptual level. The Honors sequence emphasizes proofs, so students completing it will be exempt from taking Math 109 (Mathematical Reasoning). The prerequisite is AP calculus in high school, with a 5 on the BC exam, or permission from the instructor. A grade of B- or better is necessary to continue from one course to the next in the sequence. The Honors sequence is intended for mathematics majors and prospective mathematics majors (although others are very welcome) and gives a much better view of what upper-division mathematics is like than does the standard Math 20 sequence.
The textbook for the whole Honors sequence is J. H. Hubbard and B. B. Hubbard, Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach, Third Edition (Ithaca, NY: Matrix Editions 2007). Errata have been compiled. In this course we will cover at least Chapter 1, sections 5—8; Chapter 3, sections 1—3 and 5—8; and Chapter 4, sections 1 and 5. There is a copy of the textbook on reserve at the Science and Engineering library. There are many other calculus textbooks that cover multivariable calculus, most of which cover approximately the same material (except for parts of Chapter 3) at different levels of rigor and with some differences in emphasis. Also, the Math Department has a calculus lab in AP&M B402A which is staffed by TAs from the Math 10 and 20 calculus sequences; they may be able to help with your questions when neither David nor Dan is available, but some of the material we cover may be outside the range of questions for which they are prepared.
There will be weekly homework assignments, due in section on Tuesdays, or before then in the drop box on the sixth floor of AP&M. Students are allowed to discuss the homework assignments among themselves, but are expected to turn in their own work — copying someone else's is not acceptable. Homework scores will constitute 1/6 of the final grade.
There will be two midterms, approximately in the fourth and eighth weeks of the quarter. The final is scheduled for 8:00am Wednesday 17 March 2010. Scores on the two midterms and final will constitute 1/4, 1/4 and 1/3 of the final grade, respectively. There will be no makeup tests.
| 31 Jan 10 |
deadline for applications to the Park City Math Institute Undergraduate Summer School Program |
| 4 Jan 10 |
§1.5. Limits and continuity open and closed sets convergence of sequence uniqueness of limit of sequence |
| 6 Jan 10 |
convergent sequences and closed sets subsequences limit of function example of limit HWK (due Tu 12 Jan 10). read all of §1.5 §1.5: 2, 3, 4, 8, 10, 14, 16, 20, 23 (problems 8, 10, 20, 23 may be turned in on Tu 19 Jan 10) [solutions] |
| 8 Jan 10 |
uniqueness example of no limit continuous function |
| 11 Jan 10 |
§0.5. Real numbers and series (least/greatest) upper/lower bounds every ∅ ≠ X ⊂ R that has an upper (lower) bound has least upper (greatest lower) bound convergence of series geometric series a non-decreasing series converges iff it is bounded absolute convergence §1.5. Series of vectors and matrices convergent series absolute convergence |
| 13 Jan 10 |
§0.7. Complex numbers real and imaginary parts, modulus, argument addition and multiplication §1.5. Series of vectors and matrices series defining ez converges Euler's formula for A square matrix with |A | < 1, (I - A)-1 = I + A + A2 + … the set of invertible matrices is open HWK (due Tu 19 Jan 10). §0.5: 4, 5 §0.7: 3, 4, 10, 11, 13 §1.6: 2, 6, 7 [solutions] |
| 15 Jan 10 |
Cauchy sequences (not in text) definition Cauchy sequences are bounded every Cauchy sequence in R converges §1.6. Compactness definition of bounded a non-empty set is compact if it is closed and bounded Bolzano-Weierstrass theorem: every sequence is a compact set has a convergent subsequence |
| 18 Jan 10 |
no lecture: MLK holiday |
| 20 Jan 10 |
definition of supremum, infimum, maximum (value), minimum (value) a real-valued continuous function on a compact set has a maximum and a minimum the derivative of a real-valued differentiable function vanishes at its maximum and minimum Mean Value Theorem HWK (due Tu 26 Jan 10). §1.7: 2, 4, 5, 6, 7, 10, 11, 13 [solutions] |
| 22 Jan 10 |
§1.7. Derivatives in several variables definition of derivative of real-valued function of one variable derivative as approximating linear transformation definition of partial derivatives definition of Jacobian if an approximating linear transformation exists, it is the Jacobian |
| 25 Jan 10 |
example differentiability implies continuity definition of directional derivative example computing derivative of matrix functions from definition |
| 27 Jan 10 |
Midterm 1, covering §0.5, 0.7, 1.5 (except uniform convergence), 1.6 (except Fundamental Theorem of Algebra), 1.7 |
| 29 Jan 10 |
Midterm 1 solutions Extra Credit (due M 1 Feb 10). Let X ⊂ Rn. Prove that f : X → Rm is continuous iff for all open sets U ⊂ Rm, f -1(U ) is open in X. HWK (due Tu 2 Feb 10). §1.7: 15, 16abc, 18, 19 [solutions] |
| 1 Feb 10 |
§1.8. Computing derivatives rules for multivariable derivatives the chain rule example §1.9. Criteria for differentiability example of continuous but not differentiable function |
| 3 Feb 10 |
partial derivatives are not continuous definition of continously differentiable continuously differentiable implies differentiable §2.10. Inverse and implicit function theorems definition of locally invertible Inverse Function Theorem example of calculating an approximate inverse example of sphere being locally the graph of a function HWK (due Tu 9 Feb 10). §1.7: 22 §1.8: 3, 4, 6, 9, 11 §1.9: 1, 2 §2.10: 2, 6, 8, 9 [solutions] |
| 5 Feb 10 |
Implicit Function Theorem sphere example example demonstrating that the converse of the Implicit Function Theorem is false §3.1. Manifolds definition of the graph of a multivariable function definition of smooth k dimensional manifold examples |
| 8 Feb 10 |
examples of 1 and 4 dimensional manifolds preimage of 0 under a C1 map F with DF onto is a manifold, and conversely 4 linked rods example Extra Credit (due F 12 Feb 10). Find functions defining a torus, a pair of pants, and a pair of pants where a cone at the crotch point intersects the pants in two "upward" angles and two "downward" angles. |
| 10 Feb 10 |
preimage of a manifold under a C1 map f with [Df] onto is a manifold definition of manifold is invariant under affine transformation of coordinates definition of a parameterization of a manifold examples HWK (due W 17 Feb 10). §3.1: 2, 6, 7, 10, 11, 13, 16 §3.2: 1, 5, 8, 9 [solutions] |
| 12 Feb 10 |
§3.2. Tangent spaces definition of tangent space and tangent plane example when manifold is given by graph of known function derivation of [Df(a)] = -[D1F(c) ... Dn-kF(c)]-1 [Dn-k+1F(c) ... DnF(c)] tangent plane at c is ker [DF(c)] |
| 15 Feb 10 |
no lecture: Presidents' Day holiday |
| 17 Feb 10 |
§3.3. Taylor polynomials review of single variable Taylor polynomials formula for multivariable Taylor polynomials multi-indices partial derivatives commute when they are differentiable example Extra Credit (due F 26 Feb 10). Prove that the area of any triangle with vertices at integer points in R2 is equal to i + b/2 - 1, where i and b are the numbers of integer coordinate points inside and on the triangle, respectively. Prove that the area of any polygon with vertices at integer points in R2 is equal to i + b/2 - 1. HWK (due Tu 23 Feb 10). §3.3: 2, 5, 9, 13 [solutions] |
| 19 Feb 10 |
review for Midterm 2 |
| 22 Feb 10 |
Midterm 2, covering §1.8, 1.9, 2.10, 3.1, 3.2, 3.3 |
| 24 Feb 10 |
§3.5. Quadratic forms example of degree 2 Taylor polynomial definition of quadratic form and equivalence to symmetric matrix every quadratic form can be written as a sum and difference of squares, i.e., diagonalized the number of positive and negative terms is the signature, and is independent of the diagonalization examples HWK (due Tu 2 Mar 10). §3.5: 2, 5, 8, 9, 13 §3.6: 1, 2, 5 [solutions] |
| 26 Feb 10 |
Midterm 2 solutions |
| 1 Mar 10 |
definition of positive/negative definite signature of a quadratic form is well-defined if Q is positive definite, then Q(x ) ≥ c|x |2 §3.6. Classifying critical points definition of extremum derivative vanishes at extrema in open sets definition of critical point example Extra Credit (due M 8 Mar 10). Use what we've learned about quadratic forms to prove that symmetric matrices have only real eigenvalues |
| 3 Mar 10 |
identifying critical points as maxima, minima, saddle points, or degenerate §3.7. Constrained critical points example at a local extremum of a function restricted to a manifold, the tangent space to the manifold is contained in the kernel of the derivative of the function definition of a critical point of a function constrained to a manifold examples HWK (due Tu 9 Mar 10). §3.7: 2, 3, 4, 5, 7, 13 [solutions] |
| 5 Mar 10 |
existence of Lagrange multipliers examples |
| 8 Mar 10 |
§4.1. Definition of multivariable integration definition of the characteristic function of a set definition of support of a function definition of dyadic cubes and dyadic pavings definition of integrable example properties of integrable functions the integral of a function that is the product of functions of disjoint sets of variables is the product of the integrals of those functions example |
| 10 Mar 10 |
definition of n dimensional volume volume of an interval and of a product of intervals properties of volume §4.5. Fubini's theorem calculating multivariable integrals by iterated single variable integrals examples for bounds of integration HWK (not to hand in). §4.1: 9, 11, 13, 14 §4.5: 2, 3, 4, 5, 6, 11 |
| 12 Mar 10 |
examples of evaluating integrals |
|
14 Mar 10 sunday |
review for final 11am-1pm in calculus lab, basement of AP&M |
|
17 Feb 10 wednesday |
Final, covering §0.5, 0.7, 1.5-1.9 (not uniform continuity nor Fundamental Theorem of Algebra), 2.10, 3.1-3, 3.5-7, 4.1, 4.5 Center Hall 203, 8am-11am |