ASO: Introduction to Manifolds - Material for the year 2020-2021

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Prof. Kevin McGerty
Course Term: 
Course Lecture Information: 

8 lectures

Course Overview: 

In this course, the notion of the total derivative for a function $f \colon \mathbb{R}^m \rightarrow \mathbb{R}^n$ is introduced. Roughly speaking, this is an approximation of the function near each point in $\mathbb{R}^n$ by a linear transformation. This is a key concept which pervades much of mathematics, both pure and applied. It allows us to transfer results from linear theory locally to nonlinear functions. For example, the Inverse Function Theorem tells us that if the derivative is an invertible linear mapping at a point then the function is invertible in a neighbourhood of this point. Another example is the tangent space at a point of a surface in $\mathbb{R}^3$, which is the plane that locally approximates the surface best.

Learning Outcomes: 

Students will understand the concept of derivative in n dimensions and the implict and inverse function theorems which give a bridge between suitably nondegenerate infinitesimal information about mappings and local information. They will understand the concept of manifold and see some examples such as matrix groups.

Course Synopsis: 

Definition of a derivative of a function from $\mathbb{R}^m$ to $\mathbb{R}^n$; examples; elementary properties; partial derivatives; the chain rule; the gradient of a function from $\mathbb{R}^n$ to $\mathbb{R}$; Jacobian. Continuous partial derivatives imply differentiability, Mean Value Theorems. [3 lectures]

The Inverse Function Theorem and the Implicit Function Theorem (proofs non-examinable). [2 lectures]

The definition of a submanifold of $\mathbb{R}^m$. Its tangent and normal space at a point, examples, including two-dimensional surfaces in $\mathbb{R}^3$. [2 lectures]

Lagrange multipliers. [1 lecture]

Reading List: 

Theodore Shifrin, Multivariable Mathematics (Wiley, 2005). Chapters 3-6.

T. M. Apostol, Mathematical Analysis: Modern Approach to Advanced Calculus (World Students) (Addison Wesley, 1975). Chapters 6 and 7.

S. Dineen, Multivariate Calculus and Geometry (Springer, 2001). Chapters 1-4.

J. J. Duistermaat and J A C Kolk, Multidimensional Real Analysis I, Differentiation (Cambridge University Press, 2004).

M. Spivak, Calculus on Manifolds: A modern approach to classical theorems of advanced calculus, W. A. Benjamin, Inc., New York-Amsterdam, 1965.

Further Reading: 

William R. Wade, An Introduction to Analysis (Second Edition, Prentice Hall, 2000). Chapter 11.

M. P. Do Carmo, Differential Geometry of Curves and Surfaces (Prentice Hall, 1976).

Stephen G. Krantz and Harold R. Parks, The Implicit Function Theorem: History, Theory and Applications (Birkhaeuser, 2002).