Integrated MSc-PhD Program


KSM3E05: Analysis on Manifolds


The Algebra and Topology of {\bf R}^n, Differentiation, Derivative, Continuously Differentiable Functions, The Chain Rule, The Inverse Function Theorem, The Implicit Function Theorem, The Integral over a Rectangle, Existence of the Integral, Evaluation of the Integral, The Integral over a Bounded Set, Rectifiable Sets, Improper Integrals, Partitions of Unity, The Change of Variables Theorem, Diffeomorphisms in {\bf R}^n, Proof of the Change of Variables Theorem , Application of Change of Variables, Manifolds, The Volumne of a Parallelopiped, The Volume of a Parametrized-Manifold, Manifolds in {\bf R}^n, The Boundary of a Manifold, Integrating a Scalar Function over a Manifold, Differential Forms, Multilinear Algebra, Alternating Tensors, The Wedge Product, Tangent Vectors and Differential Forms, The Differential Operator, Application to Vector and Scalar Fields, The Action of a Differentiable Map, Stokes’ Theorem, Integrating Forms over Parametrized-Manifold, Orientable Manifolds, Integrating Forms over Oriented Manifolds.

Suggested texts:

            1. Analysis on Manifolds by James Munkres
            2. Calculus on Manifolds by Michael Spivak