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, Meanvalue 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 


Midterm 

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 kforms, 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 
