everything.study
18.101

Analysis and Manifolds

3-0-9 units · Undergraduate

Introduction to the theory of manifolds: vector fields and densities on manifolds, integral calculus in the manifold setting and the manifold version of the divergence theorem. 18.901 helpful but not required.

Fall 2005 · Analysis II

Topics

  • Metric Spaces, Continuity, Limit Points
  • Compactness, Connectedness
  • Differentiation in n Dimensions
  • Conditions for Differentiability, Mean Value Theorem
  • Chain Rule, Mean-value Theorem in n Dimensions
  • Inverse Function Theorem
  • Reimann Integrals of Several Variables, Conditions for Integrability
  • Conditions for Integrability (cont.), Measure Zero
  • Fubini Theorem, Properties of Reimann Integrals
  • Integration Over More General Regions, Rectifiable Sets, Volume
  • Improper Integrals
  • Exhaustions
  • Compact Support, Partitions of Unity
  • Partitions of Unity (cont.), Exhaustions (cont.)
  • Review of Linear Algebra and Topology, Dual Spaces
  • Tensors, Pullback Operators, Alternating Tensors
  • Alternating Tensors (cont.), Redundant Tensors
  • Wedge Product
  • Determinant, Orientations of Vector Spaces
  • Tangent Spaces and k-forms, The d Operator
  • The d Operator (cont.), Pullback Operator on Exterior Forms
  • Integration with Differential Forms, Change of Variables Theorem, Sard’s Theorem
  • Poincare Theorem
  • Generalization of Poincare Lemma
  • Proper Maps and Degree
  • Proper Maps and Degree (cont.)
  • Regular Values, Degree Formula
  • Topological Invariance of Degree
  • Canonical Submersion and Immersion Theorems, Manifolds
  • Examples of Manifolds
  • Tangent Spaces of Manifolds
  • Differential Forms on Manifolds
  • Orientations of Manifolds
  • Integration on Manifolds, Degree on Manifolds
  • Degree on Manifolds (cont.), Hopf Theorem
  • Integration on Smooth Domains
  • Integration on Smooth Domains (cont.), Stokes’ Theorem

Readings

  • (M) Munkres, J. Analysis on Manifolds . Cambridge, MA: Perseus Publishing, 1991. ISBN: 0201510359, ISBN: 0201315963 (paperback).
    Listed reading
  • (S) Spivak, M. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus . Cambridge, MA: Perseus Publishing, 1965. ISBN: 0805390219.
    Listed reading
  • LEC #: 2 · TOPICS: Compactness, Connectedness · READINGS: M, sections 1 and 2
    Reading assignment
  • LEC #: 3 · TOPICS: Differentiation in n Dimensions · READINGS: M, end of section 4 on connectedness
    Reading assignment
  • LEC #: 8 · TOPICS: Reimann Integrals of Several Variables, Conditions for Integrability · READINGS: M, section 10
    Reading assignment
  • LEC #: 10 · TOPICS: Fubini Theorem, Properties of Reimann Integrals Graded · READINGS: M, section 11
    Reading assignment
  • LEC #: 13 · TOPICS: Exhaustions · READINGS: M, theorem 15.3 and following paragraph
    Reading assignment
  • LEC #: 15 · TOPICS: Partitions of Unity (cont.), Exhaustions (cont.) · READINGS: M, sections 1 and 2
    Reading assignment
  • LEC #: 22 · TOPICS: The d Operator (cont.), Pullback Operator on Exterior Forms · READINGS: M, section 29
    Reading assignment
  • LEC #: 24 · TOPICS: Poincare Theorem · READINGS: SN, section 1
    Reading assignment
  • LEC #: 34 · TOPICS: Orientations of Manifolds · READINGS: MLA, section 7
    Reading assignment
  • LEC #: 35 · TOPICS: Integration on Manifolds, Degree on Manifolds · READINGS: M, section 34, pp. 281-286 only
    Reading assignment
  • The main textbook for this course is:
    Listed reading
  • Munkres, J. Analysis on Manifolds . Cambridge, MA: Perseus Publishing, 1991. ISBN: 0201510359, ISBN: 0201315963 (paperback).
    Listed reading
  • Spivak, M. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus . Cambridge, MA: Perseus Publishing, 1965. ISBN: 0805390219.
    Listed reading

Prerequisites & course information

Subject meets with 18.1011 Prereq: ( 18.06 , 18.700 , or 18.701 ) and ( 18.100A , 18.100B , 18.100P , or 18.100Q ) U (Fall) 3-0-9 units

Sources