# Assignments

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

(S) 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 # 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, 2-7
5 Chain Rule, Mean-value 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 k-forms, 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 5-7), 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 5-14 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