This module is not currently running in 2024 to 2025.
Functions of several variables occur in many important applications. In this module we introduce the derivative for functions of several variables and derive an important consequence, namely the chain rule. We use this to calculate maxima and minima and Taylor series for functions of several variables. We also discuss the important problem of finding maxima and minima of functions subject to a constraint using the method of Lagrange multipliers. Furthermore, we define different ways to integrate functions of several variables such as arclength integrals, line integrals, surface integrals and volume integrals. Outline Syllabus includes: Continuity and Differentiation; tangent plane; swapping order of partial derivatives; implicit function theorem; inverse function theorem; paths independence of line integrals; use of polar, cylindrical and spherical polar coordinates; integral theorems such as Green's theorem.
48
80% Examination, 20% Coursework
Lang: Calculus of Several Variables, 3rd edition, Springer 1987.
Salas & Hille: Calculus - Several Variables, 7th edition, Wiley, 1995.
See the library reading list for this module (Canterbury)
The intended subject specific learning outcomes
On successful completion of this module students will:
a) have a grasp of formal definitions and rigorous proofs in functions of several variables;
b) have gained an appreciation of how to generalise concepts previously encountered in one-dimensional analysis to higher dimensions and potential difficulties;
c) be aware of applications of basic techniques and theorems of functions of several variables in other areas of mathematics, e.g., optimisation theory, mechanics.
d) be able to work with fundamental concepts in functions of several variables, such as continuity and differentiability;
e) be able to apply abstract ideas to concrete problems in analysis;
f) be able to perform calculations for specific examples involving functions of several variables.
The intended generic learning outcomes
Students who successfully complete this module will have further developed:
a) a logical, mathematical approach to solving problems;
b) their ability to communicate solutions, simple proofs and calculations;
c) their numeracy and computational skills;
d) their ability to plan and carry out effective ways of studying;
e) their ability to read and comprehend mathematical ideas.
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