Synopsis for Multivariable Calculus

Overview

In this course, the notion of the total derivative for a function is introduced. Roughly speaking, this is an approximation of the function near each point in 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 , which is the plane that locally approximates the surface best.

Synopsis

Definition of a derivative of a function from to ; examples; elementary properties; partial derivatives; the chain rule; the gradient of a function from to ; Jacobian. Continuous partial derivatives imply differentiability, Mean Value Theorems. Higher order derivatives. [3 lectures]

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

The definition of a submanifold of . Its tangent and normal space at a point, examples, including two-dimensional surfaces in . [2 lectures]

Lagrange multipliers. [1 lecture]

1. Theodore Shifrin, Multivariable Mathematics (Wiley, 2005). Chapters 3-6.
2. T. M. Apostol, Mathematical Analysis: Modern Approach to Advanced Calculus (World Students) (Addison Wesley, 1975). Chapters 12 and 13.
3. S. Dineen, Multivariate Calculus and Geometry (Springer, 2001). Chapters 1-4.
4. J. J. Duistermaat and J A C Kolk, Multidimensional Real Analysis I, Differentiation (Cambridge University Press, 2004).

Further Reading

1. William R. Wade, An Introduction to Analysis (Second Edition, Prentice Hall, 2000). Chapter 11.
2. M. P. Do Carmo, Differential Geometry of Curves and Surfaces (Prentice Hall, 1976).
3. Stephen G. Krantz and Harold R. Parks, The Implicit Function Theorem: History, Theory and Applications (Birkhaeuser, 2002).