MATH20132 - 2010/2011

 

Unit specification

Aims

The aim of this lecture course is to introduce the basic ideas of calculus of several variables.

Brief description

Functions of several variables were briefly considered in first year calculus courses when the notion of partial derivative was introduced. Although there are some similarities with the familiar theory of one real variable, the theory for functions of several variables is far richer. For example, for functions of several variables, the critical points might be maxima, minima or saddle points (which are minima in one direction and maxima in another direction). A key idea is to generalize the definition of the derivative at a point to the the derivative of a map f: Rⁿ → R^m at a point a of Rⁿ. This is the Fréchet derivative, which is a linear map df(a): Rⁿ → R^m (often represented by a matrix whose entries are partial derivatives) which gives the best approximation to the function at the point a. This derivative is used in a number of very elegant and useful results, in particular the Inverse Function Theorem and the Implicit Function Theorem, and is a key notion in the study of the critical points of functions of several variables.

The Fréchet derivative is an example of a differential 1-form on Rⁿ and so naturally leads on to an introduction to the basic ideas of differential k-forms. Differential k-forms are fundamental in the integral calculus of functions of several variables and this is briefly considered.

Intended learning outcomes

On the successful completion of this lecture students should:

• understand the notion of the limit of a function of several variables at a point and be able to find simple limits;
• understand the notion of a continuous function of several variables;
• understand the directional derivatives, the partial derivatives and the Fréchet derivative of a function of several variables at a point; be able to find these; and understand the relationship between these notions;
• be able to find the critical points on a real-valued function of several variables and determine the nature of non-degenerate critical points using the Hessian matrix;
• understand and be able the use the Chain Rule, the Inverse Function Theorem and the Implicit Function Theorem;
• be able to apply the method of Lagrange multipliers to simple extremum problems with a constraint;
• understand the notion of a differential k-form on an open subset of Rⁿ; be able to evaluate such forms at a point; be able to evaluate the wedge product of two forms and the derivative of a form; be able to evaluate line intgrals of 1-forms and surface integrals of 2-forms over a surface parametrized by a rectangle.

Future topics requiring this course unit

The ideas in this course are used in many areas of pure and applied mathematics. A natural follow on course unit is the level 3 Calculus on Manifolds.

Syllabus

1. Continuous functions of several variables.
2. Differentiation of real-valued functions of several variables.
3. Critical points and higher partial derivatives.
4. Differentiation of vector-valued functions of several variables.
5. Differential forms and integration of differential forms.

Textbooks

M.J. Field, Differential Calculus and its Applications, Van Nostrand 1976.

W. Fleming, Functions of Several Variables, Addison-Wesley 1965.

J. and B. Hubbard, Vector Calculus, Linear Algebra, and Differential Forms, Prentice Hall 1998.

C.H. Edwards, Jr., Advanced Calculus of Several Variables, Dover Publications 1994.

R. Courant and F. John, Introduction to Calculus and Analysis, Volume 2, Wiley 1974.

H.M. Edwards, Advanced Calculus: a Differential Forms Approach, Birkhauser 1994.

Learning and teaching processes

Two lectures and one tutorial class each week. Attendance at a weekly tutorial class is an essential part of the course. A weekly problem sheet will be issued and other problems will be discussed in the tutorial classes. Students should expect to spend at least four hours each week on private study for this course unit.

Assessment

A coursework test in first week after Easter (to be confirmed - Easter is very late next year so it may be before Easter): weighting 15%;

2 hours end of semester examination: weighting 85%.

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