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Lecture Notes



Lecture Notes

The lecture notes were taken by a student in the class. For all of the lecture notes, including a table of contents, download the following file (PDF - 1.6 MB).


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

 








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