Handbook 1996 : Faculty of Science (Volume 4 page 206)
Mathematics subject : Next:618-121 | Prev:618-111 | Search | Help

618-112 Mathematics 1B (Advanced)

Note: Students may gain credit for only one of 618-102 (1995 Handbook), 618-112, 618-122, 618-200 or 618-211.

Credit points: 12.5

Coordinator: Professor J H Rubinstein

Prerequisite: Mathematics 618-111 or by invitation by the Head of Department.

Contact: 39 lectures (three a week), 13 x 1-hour tutorials and 39 hours problem solving

Timetable: Second semester

Objectives:

On completion of this subject, students should :

Comprehend:

• the basic properties of sequences and series, including Taylor series for functions;
• the concepts of abstract vector spaces and inner product spaces;
• the uses and properties of linear transformations;
• the role of eigenvalues and eigenvectors in the study of such mappings;
• the fundamental ideas in the calculus of functions of several variables.

Have developed:

• an ability to use tests to decide if sequences and series converge or diverge;
• the skills to find coordinates and matrices to represent vectors and linear transformations;
• the ability to change coordinate systems to simplify problems involving vector spaces and linear transformations;
• the skills to solve problems involving contours of surfaces;
• skills to find extrema of functions and to find volumes using differentiation and integration.

Appreciate:

• the role of series in estimation of functions;
• the role of linear algebra in finding invariants and bringing out the underlying geometry in problems;
• the similarities and differences between functions of one variable and multivariate functions.

Content:

Sequences and series: convergence and divergence of sequences and series; tests for convergence; Taylor's theorem and series representation of elementary functions. Linear algebra: vector spaces in general, axioms, linear independence, basis sets, dimension, Rⁿ and Cⁿ; inner products in real and complex spaces; linear transformations, matrix of a linear transformation, change of basis; eigenvectors and eigenvalues, diagonalisation of real symmetric matrices, applications -- including applications to geometry; symmetry groups of matrices of R² and R³. Multivariable calculus: functions of several variables, level curves, heights; partial derivatives, commutation of mixed partial derivatives; total derivative, gradient vector, directional derivatives and applications; chain rule; coordinate transformations, Jacobi matrix and determinant; Hessian matrix, maxima and minima of functions of several variables; introduction to double and triple integrals.

Assessment:

Up to 26 pages of written assignments and up to three hours of end-of-semester written examination and class tests totalling not more than 1.5 hours.

Mathematics subject : Next:618-121 | Prev:618-111 | Search | Help
Handbook 1996 : Faculty of Science (Volume 4 page 206)

Status:          Official 1996
Date created:    Oct  9 1995
Last modified:   Oct  9 1995
Authorised by:   Academic Registrar
Email enquiries: Course_Information@registrar.unimelb.edu.au

Copyright © University of Melbourne 1995,1996.