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This course continues from Analysis I (18.100B), in the direction of manifolds and global analysis. The aim of this course is to cover multivariable calculus. At the end of this course, it will focus on the theory of differential forms in n-dimensional vector spaces and manifolds. Those who are interested in this field this course is the best course for them.


This course does not involve any written exams. Students need to answer 5 assignment questions to complete the course, the answers will be in the form of written work in pdf or word. Students can write the answers in their own time. Each answer needs to be 200 words (1 Page). Once the answers are submitted, the tutor will check and assess the work.


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Course Credit: MIT

Course Curriculum

Module 01
lecture1 Metric Spaces, Continuity, Limit Points 00:15:00
lecture2 Compactness, Connectedness 00:20:00
lecture3 Differentiation in n Dimensions 00:20:00
lecture4 Conditions for Differentiability, Mean Value Theorem 00:20:00
lecture5 Chain Rule, Mean-value Theorem in n Dimensions 00:20:00
lecture6 Inverse Function Theorem 00:25:00
lecture7 Inverse Function Theorem (cont.), Reimann Integrals of One Variable 00:15:00
lecture8 Reimann Integrals of Several Variables, Conditions for Integrability 00:25:00
lecture9 Conditions for Integrability (cont.), Measure Zero 00:20:00
lecture10 Fubini Theorem, Properties of Reimann Integrals 00:25:00
lecture11 Integration Over More General Regions, Rectifiable Sets, Volume 00:20:00
lecture12 Improper Integrals 00:10:00
Module 02
lecture13 Exhaustions 00:25:00
lecture14 Compact Support, Partitions of Unity 00:15:00
lecture15 Partitions of Unity (cont.), Exhaustions (cont.) 00:10:00
lecture16 Review of Linear Algebra and Topology, Dual Spaces 00:15:00
lecture17 Tensors, Pullback Operators, Alternating Tensors 00:20:00
lecture18 Alternating Tensors (cont.), Redundant Tensors 00:25:00
lecture19 Wedge Product 00:20:00
lecture20 Determinant, Orientations of Vector Spaces 00:25:00
lecture21 Tangent Spaces and k-forms, The d Operator 00:20:00
lecture22 The d Operator (cont.), Pullback Operator on Exterior Forms 00:25:00
lecture23 Integration with Differential Forms, Change of Variables Theorem, Sard’s Theorem 00:30:00
lecture24 Poincare Theorem 00:25:00
Module 03
lecture25 Generalization of Poincare Lemma 00:20:00
lecture26 Proper Maps and Degree 00:25:00
lecture27 Proper Maps and Degree (cont.) 00:25:00
lecture28 Regular Values, Degree Formula 00:20:00
lecture29 Topological Invariance of Degree 00:20:00
lecture30 Canonical Submersion and Immersion Theorems, Definition of Manifold 00:15:00
lecture31 Examples of Manifolds 00:20:00
lecture32 Tangent Spaces of Manifolds 00:20:00
lecture33 Differential Forms on Manifolds 00:20:00
lecture34 Orientations of Manifolds 00:20:00
lecture35 Integration on Manifolds, Degree on Manifolds 00:30:00
lecture36 Degree on Manifolds (cont.), Hopf Theorem 00:20:00
lecture37 Integration on Smooth Domains 00:20:00
lecture38 Integration on Smooth Domains (cont.), Stokes’ Theorem 00:15:00
Submit Your Assignment 00:00:00
Certification 00:00:00

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