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Created by:

Massachusetts Institute of Technology

Last updated:

December 24, 2022

Duration:

Unlimited Duration

FREE

TAKE THIS COURSE

This course includes:

Unlimited Duration

Badge on Completion

Certificate of completion

Unlimited Duration

Description

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

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Massachusetts Institute of Technology