
MAT 322 Syllabus
Analysis in Several Dimensions
Spring 2016

The revised
schedule for topics covered in lecture is as follows. You must
do
the assigned reading prior to lecture. This will make the lectures
more effective for you.
 January 26
Section 1. Review of Linear Algebra.
Section 2. Matrix Inversion and Determinants.
 January 28
Section 3. Review of Topology in Real Euclidean Space.
 February 2
Section 4. Compact Subspaces and Connected Subspaces of Real Euclidean Space.
 February 4
Section 5. The Derivative.
Problem Set 1
due in lecture.
 February 9
Section 6. Continuously Differentiable Functions.
Section 7. The Chain Rule.
 February 11
Section 8. The Inverse Function Theorem.
Problem Set 2
due in lecture.
 February 16
Section 9. The Implicit Function Theorem.
 February 18
Section 10. The Integral over a Rectangle.
Problem Set 3
due in lecture.
 February 23
Section 11. Existence of the Integral.
Section 12. Evaluation of the Integral.
 February 25
Section 13. The Integral over a Bounded Set.
Section 14. Rectifiable Sets.
Section 15. Improper Integrals.
Problem Set 4
due in lecture.
 March 1
Section 16. Partitions of Unity.
Section 17. The Change of Variables Theorem.
Section 18. Diffeomorphisms in Real Euclidean Space.
 March 3
Section 19. Proof of the Change of Variables Theorem.
Section 20. Applications of Change of Variables.
Problem Set 5
due in lecture. Practice for Exam 2.
 March 8
Review for Midterm 1.
 March 10
MIDTERM 1
No Problem Set Due This Week.
 March 22
Section 21. The Volume of a Parallelopiped.
Section 22. The Volume of a Parameterized Manifold.
 March 24
Section 23. Manifolds in Real Euclidean Space.
Problem Set 6
due in lecture.
 March 29
Section 24. The Boundary of a Manifold.
Section 25. Integrating a Scalar Function over a Manifold.
 March 31
Section 26. Multilinear Algebra.
Problem Set 7
due in lecture.
 April 5
Section 27. Alternating Tensors.
Section 28. The Wedge Product.
 April 7
Section 29. Tangent Vctors and Differential Forms.
Problem Set 8
due in lecture.
 April 12
Section 30. The Differential Operator.
 April 14
Section 31. Application to Vector and Scalar Fields.
Section 32. The action of a Differentiable Map.
Problem Set 9
due in lecture.
 April 19
Review for Midterm 2.
 April 21
MIDTERM 2
No Problem Set Due This Week.
 April 26
Section 33. Integrating Forms over Parameterized Manifolds.
Section 34. Orientable Manifolds.
 April 28
Section 35. Integrating Forms over Oriented Manfiolds.
Section 37. The Generalized Stokes's Theorem.
Problem Set 10
due in lecture.
 May 3
Section 38. Applications to Vector Analysis.
Section 39. The Poincaré Lemma.
Section 40. The deRham Groups of Punctured Euclidean Spaces.
 May 5
FINAL REVIEW
Problem Set 11
due in lecture.
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Jason Starr
4108 Math Tower
Department of Mathematics
Stony Brook University
Stony Brook, NY 117943651
Phone: 6316328270
Fax: 6316327631
Jason Starr