MT3244 - Calculus on Manifolds
MT3244 - Calculus on Manifolds
Instructor : Dr. Sagar Kalane
Schedule : Monday, Tuesday and Thursday, Time: 12 - 1 PM
Evaluation : Mid-sem - 30%, End-sem - 40%, Others- Assignments: 10%, Two Tests: 20%
Prerequisite : Linear Algebra, Real analysis 2 and Point set Topology etc.(Undergraduate level courses)
Text books and reference material :
Calculus on Manifolds : M. Spivak,
Analysis on Manifolds: J. R. Munkres.
Weekly topics covered :
3 January 23 - Introduction, Differentiation of functions of several variables, Directional derivatives, examples.
4 January 23 - Partial Derivatives, Jacobian Matrix, Theorems.
6 January 23 - Continuously differentiable functions, mixed partial derivatives, higher order partial derivatives.
10 January 23 - Exercises, Chain Rule.
12 January 23 - Applications of Chain rule.
16/ 17 January 23 - Inverse function theorem.
23 January 23 - Proof of Inverse function theorem, Exercises.
24 January 23 - Implicit function theorem.
28 January 23- Exercises on IFT, Integration.
30 January 23 - Integrable function and Examples.
31 January 23 - Measure zero, Content zero set and Integrability of discontinuous functions.
2 February 23 - Fubini's Theorem.
9 February 23 - Partitions of Unity.
11 February 23 - Proof of Partition of unity and Integrability
14 February 23 - Tutorial
16 February 23 - Change of variables
17 February 23 - All the best !!
27 February 23 - Remaining proof of change of variable and k- tensors.
28 February 23 - Properties and dimension of set of all k- tensors.
2 March 23 - Alternating k- tensors and its dimension.
6 March 23 - Volume element, Cross product and Orientation.
9 March 2023 - Tangent space, Vector field and differential form.
14 March 2023 - Properties of differential form.
16 March 2023 - Problems and properties of differential operator.
20 March 2023 - Quiz 2 and properties of Pullback.
21 March 2023 - n-chain, boundary of chain.
23 March 2023 - Integration on chains.
27 March 2023 - Proof of Steoke's Theorem on chains.
28 March 2023 - Remaining proof and Tutorial.
30 March 2023 - Definition and example of manifolds.
3 April 2023 - Characterization of manifolds.
4 April 2023 - K- form and and Tangent space of Manifold.
7 April 2023 - Orientation on Manifold.
11 April 2023 - Tutorial on Manifolds.
12 April 2023 - Quiz 3 and Stokes Theorem on Manifolds.