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Last updated:

December 24, 2022


Unlimited Duration


This course includes:

Unlimited Duration

Badge on Completion

Certificate of completion

Unlimited Duration


This course continues from Analysis I (18.100B), in the direction of manifolds and global analysis.

The first half of the course covers multivariable calculus. The rest of the course covers the theory of differential forms in n-dimensional vector spaces and manifolds.

Course Curriculum

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

About the instructor

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Massachusetts Institute of Technology
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