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Assignments



Assignments

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There are two types of assignments given in this course: daily assignments and graded assignments. The daily assignments are not graded, but one problem from each day is usually included in a graded assignment. Not all lectures have assigned daily problems.

Assignments listed in the table below are from the following textbooks and notes:

(M) Amazon logo Munkres, J. Analysis on Manifolds. Cambridge, MA: Perseus Publishing, 1991. ISBN: 0201510359, Amazon logo ISBN: 0201315963 (paperback).

(S) Amazon logo Spivak, M. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. Cambridge, MA: Perseus Publishing, 1965. ISBN: 0805390219.

(MLA) Notes on Multi-linear Algebra (PDF)

(SN) Supplementary Notes (PDF)


Lec #TopicsDAILY ASSIGNMENTSGRADED ASSIGNMENTS
1Metric Spaces, Continuity, Limit PointsM, section 3: 2, 3, 4, 8, 9
2Compactness, ConnectednessM, section 4: 1, 2, 3, 4, concentrate on 3
3Differentiation in n DimensionsM, section 5: 2, 3, 4, 5, 7
4Conditions for Differentiability, Mean Value TheoremM, section 6: 2, 5, 9, 10M, 4.3, 5.3, 6.10, 8.4

S, 2-7
5Chain Rule, Mean-value Theorem in n DimensionsM, section 7: 1, 2, 3
6Inverse Function TheoremM, section 8: 1, 2
7Inverse Function Theorem (cont.), Reimann Integrals of One VariableM, section 8: 3, 4, 5
8Reimann Integrals of Several Variables, Conditions for IntegrabilityM, section 10: 1, 3, 4, 5
9Conditions for Integrability (cont.), Measure ZeroM, section 12: 1, 2, 3, 4
10Fubini Theorem, Properties of Reimann IntegralsM, section 13: 1, 2, 4, 5M, 12.2, 13.2, 14.8, 15.4, 16.3
11Integration Over More General Regions, Rectifiable Sets, VolumeM, section 14: 1, 4, 5, 7 (Hint: look at Example 1 of section 14 for help with two of the homework problems.)
12Improper IntegralsM, section 15: 1, 2, 4, 5
13Exhaustions
Midterm
14Compact Support, Partitions of UnityM, section 16: 2, 3
15Partitions of Unity (cont.), Exhaustions (cont.)
16Review of Linear Algebra and Topology, Dual SpacesMLA, section 2: 1, 2, 3, 4
17Tensors, Pullback Operators, Alternating TensorsMLA, section 3: 1, 2, 4, 6, 7
18Alternating Tensors (cont.), Redundant TensorsMLA, section 4: 1, 2, 3, 4, 5
19Wedge ProductMLA, section 5: 1, 2 and section 6: 1
20Determinant, Orientations of Vector SpacesMLA, section 6: 2, 3, 4, 5(PDF)
21Tangent Spaces and k-forms, The d Operator
22The d Operator (cont.), Pullback Operator on Exterior FormsM, section 30: 2, 3, 4, 6
23Integration with Differential Forms, Change of Variables Theorem, Sard's TheoremSN, section 1: 1, 2, 4, 5
24Poincare TheoremSN, section 2: 1, 2, 3
25Generalization of Poincare Lemma
26Proper Maps and DegreeSN, section 4: 3, 4, 5, 6, 7
27Proper Maps and Degree (cont.)
28Regular Values, Degree Formula
29Topological Invariance of DegreeM, section 24: 6

SN, section 2: 2, section 4: 8 (need 5-7), section 6: 6
30Canonical Submersion and Immersion Theorems, Definition of ManifoldProve the canonical submersion and immersion theorems for linear maps (as stated in class).
31Examples of ManifoldsM, section 23: 1, 4, 5 and section 24: 5, 6
32Tangent Spaces of ManifoldsM, section 29: 1, 2, 3, 5
33Differential Forms on ManifoldsMLA, section 7: 1, 4, 5, 6
34Orientations of ManifoldsM, section 34: 3, 6

S, problems 5-14 on p. 120
35Integration on Manifolds, Degree on Manifolds
36Degree on Manifolds (cont.), Hopf Theorem(PDF)
37Integration on Smooth Domains(PDF)
38Integration on Smooth Domains (cont.), Stokes’ Theorem
Final Exam

 








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