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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.
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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.
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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.
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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.
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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.
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