1 |
Metric Spaces, Continuity, Limit Points |
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2 |
Compactness, Connectedness |
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3 |
Differentiation in n Dimensions |
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4 |
Conditions for Differentiability, Mean Value Theorem |
Graded assignment 1 out |
5 |
Chain Rule, Mean-value Theorem in n Dimensions |
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6 |
Inverse Function Theorem |
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7 |
Inverse Function Theorem |
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8 |
Reimann Integrals of Several Variables, Conditions for Integrability |
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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 |
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12 |
Improper Integrals |
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13 |
Exhaustions |
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Midterm |
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14 |
Compact Support, Partitions of Unity |
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15 |
Partitions of Unity (cont.), Exhaustions (cont.) |
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16 |
Review of Linear Algebra and Topology, Dual Spaces |
Graded assignment 2 due |
17 |
Tensors, Pullback Operators, Alternating Tensors |
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18 |
Alternating Tensors (cont.), Redundant Tensors |
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19 |
Wedge Product |
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20 |
Determinant, Orientations of Vector Spaces |
Graded assignment 3 out |
21 |
Tangent Spaces and k-forms, The d Operator |
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22 |
The d Operator (cont.), Pullback Operator on Exterior Forms |
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23 |
Integration with Differential Forms, Change of Variables Theorem, Sard's Theorem |
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24 |
Poincare Theorem |
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25 |
Generalization of Poincare Lemma |
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26 |
Proper Maps and Degree |
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27 |
Proper Maps and Degree (cont.) |
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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 |
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31 |
Examples of Manifolds |
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32 |
Tangent Spaces of Manifolds |
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33 |
Differential Forms on Manifolds |
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34 |
Orientations of Manifolds |
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35 |
Integration on Manifolds, Degree on Manifolds |
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36 |
Degree on Manifolds (cont.), Hopf Theorem |
Graded assignment 4 due |
37 |
Integration on Smooth Domains |
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38 |
Integration on Smooth Domains (cont.), Stokes’ Theorem |
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Final Exam |
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