Department of Mathematics

Van Vleck Hall, 480 Lincoln Drive, Madison, WI

Math 621: Analysis III

Background and Goals: 

The sequence Math 521-522-621 has the objective of conveying an understanding of the structure of analysis in itself as well as its role as a tool for other disciplines. This sequence is highly recommended for any math major and it is essential for students preparing for graduate studies in mathematics; also it should be taken by students in physics and engineering who intend to do graduate work in their areas.

Subsequent Courses: 

Graduate courses in analysis, geometry and topology

Course Content: 
  • Review of differential calculus in normed spaces.
  • Integration in Euclidean spaces.
    • Basic definitions
    • Measure and content zero, and characterization of Riemann integrability (optional).
    • Fubini's theorem.
    • Partition of unity.
    • Changes of variables.
  • Some multilinear algebra.
    • Review of determinants.
    • Multilinear maps, tensors, alternating tensors, wedge product.
  • Manifolds
    • Basic concept of a manifold, definitions of tangent space.
    • Fields and forms on manifolds, orientation.
  • Integration
    • Stokes' theorem on manifolds.
    • Euclidean measure for submanifolds.
    • Grad, curl, and div, the Laplacian.
    • The classical theorems in vector analysis by Green, Gauss and Stokes.
    • Cauchy's integral theorem and formula.
  • Other optional topcs.
credits: 
3 (N-A)
semester: 
FallSpring
prereqs: 
Math 522, or consent of instructor.

UW-Madison Department of Mathematics
Van Vleck Hall
480 Lincoln Drive
Madison, Wi  53706

(608) 263-3054

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