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Swayam

An Introduction to smooth Manifolds

Indian Institute of Technology Bangalore , NPTEL and Indian Institute of Science Bangalore via Swayam

Overview

The goal of this course is to introduce the student to the basics of smooth manifold theory. The course will start with a brief outline of the prerequisites from topology and multi-variable calculus.After the introducion of differentiable manifolds, a large class of examples, including Lie groups, will be presented. The course will culminate with a proof of Stokes' theorem on manifolds.

INTENDED AUDIENCE : Masters and PhD students in mathematics, physics, robotics and control theory, information theory and climate sciences.PREREQUISITES : Real analysis, linear algebra and multi-variable calculus, topology.INDUSTRY SUPPORT : Nil

Syllabus

COURSE LAYOUT

Week 1 : Review of topology and multi-variable calculusWeek 2 : Definition and examples of smooth manifoldsWeek 3 : Smooth maps between manifolds, submanifoldsWeek 4 : Tangent spaces and vector fieldsWeek 5 : Lie brackets and Frobenius theoremWeek 6 : Lie groups and Lie algebrasWeek 7 : Tensors and differential formsWeek 8 : Exterior derivativeWeek 9 : OrientationWeek 10 : Manifolds with boundaryWeek 11 : Integration on manifoldsWeek 12 : Stokes Theorem

Taught by

Prof. Harish Seshadri

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