# Theoretical multivariable calculus (Math 25B)

Undergraduate course, *Harvard University*, 2019

## Summary

This course is focused on the study of functions of several variables, a topic broadly referred to as mathematical analysis. This will include differentiation, integration, and rigorous proofs of the classical theorems of calculus in a broad setting. One of the central goals of the course is to introduce you to proof-writing and abstract mathematical thinking.

## Textbooks

- Pugh,
*Real mathematical analysis* - Munkres,
*Analysis on manifolds* - Spivak,
*Calculus on manifolds* - Hubbard-Hubbard,
*Vector calculus, linear algebra, and differential forms*

## Topic schedule

- Week 1: Pugh 1.2, 1.3, 1.6
- Mon. Lecture 1: Continuity, the real numbers, least upper bound property
- Wed. Lecture 2: Skeleton of calculus, continuity theorems, Dedekind cuts
- Fri.

- Week 2: Pugh 2.1, 2.2, 2.4
- Mon. Lecture 3: Convergence, sequences, limits, continuity
- Wed. Lecture 4: Topology, open/closed sets, compactness/Heine-Borel, continuity
- Fri. Homework 1 due.

- Week 3: Pugh 2.4, 3.1
- Mon. Lecture 5: More topology, interior/exterior/boundary, compactness and coverings, the Cantor set
- Wed. Lecture 6: Differentiability, mean value theorem, Taylor polynomials
- Fri. Homework 2 due.

- Week 4 : Pugh 3.1
- Mon. President’s Day (no class)
- Wed. Lecture 7: Polynomial approximation, second derivative test
- Fri. Homework 3 due.

- Week 5: Pugh 3.2
- Mon. Lecture 8: Integrability, Riemann integral
- Wed. Lecture 9: Continuous implies integrable, Fundamental theorem of calculus, measure
- Fri. Midterm 1 due at 11:59pm

- Week 6: Pugh 3.2, 4.1, 4.5
- Mon. Lecture 10: Integrability and measure, Cavalieri and Fubini
- Wed. Lecture 11: Function spaces, ODEs
- Fri. Homework 4 due, extra credit due (LUB property, Cantor’s construction of R, compactness)

- Week 7: Pugh 4.3, 4.5
- Mon. Lecture 12: Function convergence, equicontinuity, Arzela-Ascoli theorem
- Wed. Lecture 13: Arzela-Ascoli theorem, ODE existence theorem
- Fri. Homework 5 due.

Week 8: Spring break

- Week 9: Pugh 5.1, 5.2, 5.3
- Mon. Lecture 14: Multivariable derivative, chain rule
- Wed. Lecture 15: Continuous partials theorem, multivariable MVT, higher derivatives
- Fri. Homework 6 due.

- Week 10: Pugh 5.5, 5.6
- Mon. Lecture 16: Implicit function theorem
- Wed. Lecture 17: Implicit and inverse function theorems
- Fri. Homework 7 due.

- Week 11: Pugh 5.6, 5.7
- Mon. Lecture 18: manifolds and Lagrange multipliers
- Wed. Lecture 19: Stokes’ theorem, Linear and differential forms
- Fri. Midterm 2
- Sun: extra credit due (Taylor polynomials, log/exp, pictogram)

- Week 12:
- Mon. Lecture 20: integration of differential forms
- Wed. Lecture 21: exterior derivative and pullbacks
- Fri. Homework 8 due.

- Week 13:
- Mon. Lecture 22: chains and boundaries
- Wed. Lecture 23: Stokes’ theorem proof, Brouwer fixed point theorem
- Fri. Homework 9 due.

- Week 14:
- Mon. Lecture 24: Stokes’ applications: Green’s theorem, FTA
- Wed. Lecture 25: Stokes’ applications: FTA, planimeters
- Fri. HW10 due.

## Homework assignments

HW1 HW2 HW3 HW4 HW5 HW6 HW7 HW8 HW9

## Lecture notes

A pdf of lecture notes taken by Beckham Myers (course assistant).