Course Goals

150B Catalog Description:  Calculus of functions of several variables, inverse function theorem. Further topics may include exterior differential forms, Stokes’ theorem, manifolds, Sard’s theorem, elements of differential topology, singularities of maps, catastrophes, further topics in differential geometry, topics in geometry of physics. Prerequisites: MATH 150A or consent of instructor.  

1. Chapter 1. Multilinear Algebra
   
   This chapter contains the main linear algebra needed to understand integration over manifolds and related forms of integration by parts in arbitrary dimensions, i.e. Stoke's Theorem. The main point is to see that  "signed volumes" are related to alternating multi-linear functions.
    

2.  Differential Forms
   
   In this chapter we study differential forms on Euclidean spaces. A differential form is a function taking values in the space of multi-lilnear forms which we will have covered in Chapter 1.
   

3.  Integration of Forms
   
   In this chapter we will begin to realize the fruits of our efforts by showing how to integrate differential forms. A key result in this chapter is the (generalized) multi-dimensional change of variables theorem.
   

4.  Manifolds & Forms on Manifold
   
   Here we introduce the notion of a general manifold examples of which are surfaces your studied in Math 150A. Loosely speaking, a manifolds is something that locally looks like Euclidean space. We will give a number of examples and explain how to do calculus in this more general setting.  In particular, we will discuss Stoke's theorem, which is a multi-dimensional form of integration by parts.