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