Lec #  Topics  DAILY ASSIGNMENTS  GRADED ASSIGNMENTS 

1  Metric Spaces, Continuity, Limit Points  M, section 3: 2, 3, 4, 8, 9  
2  Compactness, Connectedness  M, section 4: 1, 2, 3, 4, concentrate on 3  
3  Differentiation in n Dimensions  M, section 5: 2, 3, 4, 5, 7  
4  Conditions for Differentiability, Mean Value Theorem  M, section 6: 2, 5, 9, 10  M, 4.3, 5.3, 6.10, 8.4 S, 27 
5  Chain Rule, Meanvalue Theorem in n Dimensions  M, section 7: 1, 2, 3  
6  Inverse Function Theorem  M, section 8: 1, 2  
7  Inverse Function Theorem (cont.), Reimann Integrals of One Variable  M, section 8: 3, 4, 5  
8  Reimann Integrals of Several Variables, Conditions for Integrability  M, section 10: 1, 3, 4, 5  
9  Conditions for Integrability (cont.), Measure Zero  M, section 12: 1, 2, 3, 4  
10  Fubini Theorem, Properties of Reimann Integrals  M, section 13: 1, 2, 4, 5  M, 12.2, 13.2, 14.8, 15.4, 16.3 
11  Integration Over More General Regions, Rectifiable Sets, Volume  M, section 14: 1, 4, 5, 7 (Hint: look at Example 1 of section 14 for help with two of the homework problems.)  
12  Improper Integrals  M, section 15: 1, 2, 4, 5  
13  Exhaustions  
Midterm  
14  Compact Support, Partitions of Unity  M, section 16: 2, 3  
15  Partitions of Unity (cont.), Exhaustions (cont.)  
16  Review of Linear Algebra and Topology, Dual Spaces  MLA, section 2: 1, 2, 3, 4  
17  Tensors, Pullback Operators, Alternating Tensors  MLA, section 3: 1, 2, 4, 6, 7  
18  Alternating Tensors (cont.), Redundant Tensors  MLA, section 4: 1, 2, 3, 4, 5  
19  Wedge Product  MLA, section 5: 1, 2 and section 6: 1  
20  Determinant, Orientations of Vector Spaces  MLA, section 6: 2, 3, 4, 5  (PDF) 
21  Tangent Spaces and kforms, The d Operator  
22  The d Operator (cont.), Pullback Operator on Exterior Forms  M, section 30: 2, 3, 4, 6  
23  Integration with Differential Forms, Change of Variables Theorem, Sard's Theorem  SN, section 1: 1, 2, 4, 5  
24  Poincare Theorem  SN, section 2: 1, 2, 3  
25  Generalization of Poincare Lemma  
26  Proper Maps and Degree  SN, section 4: 3, 4, 5, 6, 7  
27  Proper Maps and Degree (cont.)  
28  Regular Values, Degree Formula  
29  Topological Invariance of Degree  M, section 24: 6 SN, section 2: 2, section 4: 8 (need 57), section 6: 6 

30  Canonical Submersion and Immersion Theorems, Definition of Manifold  Prove the canonical submersion and immersion theorems for linear maps (as stated in class).  
31  Examples of Manifolds  M, section 23: 1, 4, 5 and section 24: 5, 6  
32  Tangent Spaces of Manifolds  M, section 29: 1, 2, 3, 5  
33  Differential Forms on Manifolds  MLA, section 7: 1, 4, 5, 6  
34  Orientations of Manifolds  M, section 34: 3, 6 S, problems 514 on p. 120 

35  Integration on Manifolds, Degree on Manifolds  
36  Degree on Manifolds (cont.), Hopf Theorem  (PDF)  
37  Integration on Smooth Domains  (PDF)  
38  Integration on Smooth Domains (cont.), Stokes’ Theorem  
Final Exam 