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This course continues from Analysis I (18.100B), in the direction of manifolds and global analysis.
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English
English [CC]
FREE
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Description
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 content
- 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
Instructor
Massachusetts Institute of Technology
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