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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.

**Assessment**

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.

**Certification**

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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 | ||

Assessment | |||

Submit Your Assignment | 00:00:00 | ||

Certification | 00:00:00 |

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