Content: There are many situations in pure and applied mathematics where one has to consider the continuity and differentiability of a function f: Rⁿ → R^m, such as the determinant of an n × n matrix as a function of its entries, or the wind velocity as a function of space and time. It turns out that partial derivatives, while easy to calculate, are not robust enough to yield a satisfactory differentiation theory.

The central object of study in this module is the Fréchet derivative: the derivative of f at a point x ∈ Rⁿ is interpreted as a linear transformation df(x): Rⁿ → R^m, or m × n matrix. This module establishes the basic properties of this derivative, which generalise those of single-variable calculus: the usual algebraic rules for differentiation hold true, as do appropriate versions of the chain rule, mean value theorem, Taylor's theorem, and the use of the derivative to find local minima and maxima of a real-valued function. Highlights of the module include the statement and proof of the inverse and implicit function theorems, which have many applications in both geometry and the study of solutions of nonlinear equations, and the Lagrange multiplier theorem for the minimization/maximization of constrained functions.

We will also study norms on infinite-dimensional vector spaces and some applications.

Aims:

1. To develop the theory of the Fréchet derivative as a linear map and study its relationship with the Jacobian matrix of partial derivatives.
2. To extend the results on differentiation of real-valued functions of a single variable to functions between higher-dimensional linear spaces.
3. To introduce the basic theory of normed vector spaces as needed for this theory and to provide a basis for later modules.
4. To show how different branches of mathematics, in this instance linear algebra and analysis, combine to give an aesthetically satisfying and powerful theory.
5. To encourage self-motivated study of mathematics, through examples sheets and further reading.

Objectives: By the end of this module students should have a basic working knowledge of higher-dimensional calculus. Students should understand this in the context of normed spaces and appreciate the role this level of abstraction plays in the theory. They should understand basic linear functional analysis to the extent of being able to follow it up in the relevant third year modules. They should also be in a position to make use of more advanced textbooks if they wish to go further into these theories.

Some related books:

1. R. Abraham, J. E. Marsden, T. Ratiu. Manifolds, Tensor Analysis, and Applications. Springer, second edition, 1988.
2. T. M. Apostol. Mathematical Analysis. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., second edition, 1974.
3. R. Coleman. Calculus on normed vector spaces, Springer 2012. [available online via Warwick's library]
4. J. J. Duistermaat, J. A. C. Kolk. Multidimensional Real Analysis I : Differentiation, CUP, 2004 [available online via Warwick's library]
5. T. W. Körner. A Companion to Analysis: A Second First and First Second Course in Analysis, volume 62 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2004.
6. J. E. Marsden and A. Tromba. Vector Calculus. Macmillan Higher Education, sixth edition, 2011.
7. J. R. Munkres. Analysis on Manifolds. Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1991.
8. W. Rudin. Principles of Mathematical Analysis. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York-Auckland-Düsseldorf, third edition, 1976.
9. M. Spivak. Calculus on Manifolds. A Modern Approach to Classical Theorems of Advanced Calculus. W. A. Benjamin, Inc., New York-Amsterdam, 1965.