General information

Instructor: Nicolas Broutin, Room 1149-4
Time and Location:
  • Lectures: Mon and Wed from 8:15 to 9:30am in 202.
  • Recitations: Fri from 9:45 to 11:00am in 202.
Office hours: Mon-Wed 10:00--11:00am and Fri 11:00--12:00.
Textbook: Rudin, Foundations of Mathematical Analysis, 3rd Edition, McGraw-Hill, 1974.

Course Description

The course covers the essential tools and notions in multivariate analysis: Differentiation (multivariable), Implicit function theorem; Riemann integral (multivariable); Vector fields and differential forms, Stokes’ theorem; Metric notions (including area, volume); Introduction to Lebesgue theory.

Annotated Material

The following pages contain the results that we see during the course. The proofs omitted, but there are many comments, and a number of typical exercises that one should be able to solve to pass.

A. Differentiation
  1. Essential Notions and Concepts
  2. Inverse and Implicit Functions Theorems
  3. Around Exchanges of limits
    1. Higher order derivatives
    2. Derivatives of parametric integrals
B. Integration (still in progress)
  1. Integration of functions on $\mathbb R^k$
  2. Differential Forms
    1. Definitions
    2. Computation rules
    3. Differentiation
    4. Change of variables
  3. Simplices and Chains
  4. Stokes Theorem, closed and exact forms


  • J. Munkres. Analysis on Manifolds. Addison-Wesley, Reading, 1991.
  • W. Rudin. Principles of Mathematical Analysis. 3rd Edition, McGraw-Hill, 1974.
  • M. Spivak. Calculus on Manifolds. Addison-Wesley, Readin, 1995.
  • W. Wade. Introduction to Analysis. 4th Edition, Pearson, 2009.