General Description
Functions of several variables were briefly considered in the first year core unit MT1222.  Although there are some similarities with the familiar theory of one real variable, the theory for functions of several variables if 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 are maxima in another direction).   A key idea is to generalize the definition of the derivative to the case of maps f:Rⁿ --> R^m .  This is the Fréchet derivative, which is a linear map Df: Rⁿ --> R^m (often represented by a matrix) which gives the best approximation to the function.  This course deals with a number of very elegant and useful results, including the Inverse Function Theorem and Implicit Function Theorem.

Aims
This course unit aims to give an introduction to  functions in several variables, with an emphasis on ideas and examples .

Learning Outcomes
On successful completion of the course unit students will be able to differentiate functions of several variables and to analyse their critical points.  They will know the statements of  Sard's theorem, The mean value theorem, The inverse and implicit function theorems and Taylor's theorem, and have some familiarity with their proofs.  Students will have seen the proofs of the main results, but will not be required to have an intimate knowledge of the details.

Prerequisites
211 (ex-UMIST), MT2222 (ex-VUM)).

Future topics requiring this course unit
This course will be useful for MATH40111, MATH40512  

Content
Continuity of functions between Euclidean spaces Rⁿ.  Open set in Rⁿ.  Revision of finite dimensional vector spaces.   [2]
Partial derivatives. Frechet derivatives of functions.  Critical points. Maxima and minima. Sard's theorem. The chain rule. [6]
The mean value theorem for functions of several variables. [2]
The implicit function theorem.  The inverse function theorem . Applications of Implicit function theorem. [8]
Higher derivatives.  Taylor's theorem for functions of several variables. [4]
Preview of infinite dimensional analysis. [2]

Teaching and learning methods
Two lectures each week and weekly examples classes.
 
 

Learning hours

Activity	Hours
Staff/student contact	36
Private study	60
Total hours	96

 

Assessment

Activity	Length	Weighted within unit
End of semester examination	2 hours	100%

Core learning materials
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.

Printed notes will be distributed.