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Analysis II >> Content Detail



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Lec #TopicsKEY DATES
1Metric Spaces, Continuity, Limit Points
2Compactness, Connectedness
3Differentiation in n Dimensions
4Conditions for Differentiability, Mean Value TheoremGraded assignment 1 out
5Chain Rule, Mean-value Theorem in n Dimensions
6Inverse Function Theorem
7Inverse Function Theorem
8Reimann Integrals of Several Variables, Conditions for Integrability
9Conditions for Integrability (cont.), Measure ZeroGraded assignment 1 due 2 days after Lec #9
10Fubini Theorem, Properties of Reimann IntegralsGraded assignment 2 out
11Integration Over More General Regions, Rectifiable Sets, Volume
12Improper Integrals
13Exhaustions
Midterm
14Compact Support, Partitions of Unity
15Partitions of Unity (cont.), Exhaustions (cont.)
16Review of Linear Algebra and Topology, Dual SpacesGraded assignment 2 due
17Tensors, Pullback Operators, Alternating Tensors
18Alternating Tensors (cont.), Redundant Tensors
19Wedge Product
20Determinant, Orientations of Vector SpacesGraded assignment 3 out
21Tangent Spaces and k-forms, The d Operator
22The d Operator (cont.), Pullback Operator on Exterior Forms
23Integration with Differential Forms, Change of Variables Theorem, Sard's Theorem
24Poincare Theorem
25Generalization of Poincare Lemma
26Proper Maps and Degree
27Proper Maps and Degree (cont.)
28Regular Values, Degree FormulaGraded assignment 3 due
29Topological Invariance of DegreeGraded assignment 4 out
30Canonical Submersion and Immersion Theorems, Manifolds
31Examples of Manifolds
32Tangent Spaces of Manifolds
33Differential Forms on Manifolds
34Orientations of Manifolds
35Integration on Manifolds, Degree on Manifolds
36Degree on Manifolds (cont.), Hopf TheoremGraded assignment 4 due
37Integration on Smooth Domains
38Integration on Smooth Domains (cont.), Stokes’ Theorem
Final Exam

 








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