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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