| Instructor: |
Maxim Braverman |
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457 LA, ext. 8769 |
| Class hours: |
MW 5:50-7:20 PM |
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Location:
511 LA |
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Textbook: |
I will use different books for different parts of the course
and will distribute printouts after each class. A big part of the course
will follow the book
"Mathematical
Analysis" by Andrew Browder |
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Prerequisites: |
Analysis 1, Linear Algebra. |
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Grading: |
Weekly homework problems
may be done collaboratively. The take-home final exam must be entirely your
own work. |
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Course Description: |
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In the first part of the course I will discuss some elements
of multivariable integration theory on Rn and on
manifolds. In particular I will introduce differential forms and prove the
general Stockes' theorem. Then I will discuss elements of vector analysis
and show that classical
Green’s, Gauss-Ostrogradski’s,
and Stockes’ Theorems a particular cases of the general Stockes' theorem for
differential forms.
In the second part of the course I will present the basic theory of
functions of one complex variable. In particular, the methods of computing
integrals using complex analysis will be presented.
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Main topics to be covered: |
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Introduction to
Manifolds.
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Tensor Algebra, Exterior
Algebra. Tensor and Vector Fields on Manifolds.
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Stockes’ Theorem for
Differential Forms and its consequences (classical Green’s, Gauss-Ostrogradski’s,
and Stockes’ Theorems)
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Calculus of Vector
Fields on Manifolds. Harmonic functions.
- Functions of one complex variables. Holomorphic functions.
- Cauchy integral formula. First properties of holomorphic functions.
- Classification of singular points of complex analytic functions.
- The residue formula. Computing of integrals using complex analysis.
- Analytic continuation. Computing integrals using multivalued analytic
functions.
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| Homework: |
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