This course continues from Analysis I (18.100B), in the direction of manifolds and global analysis.
FREE
This course includes
Hours of videos
1055 years, 5 months
Units & Quizzes
38
Unlimited Lifetime access
Access on mobile app
Certificate of Completion
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 Currilcum
- 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
