Handbook 1996 : Faculty of Engineering (Volume 4 page 86)
First Year Engineering subject : Next:618-181 | Prev:618-171 | Search | Help

1. First Year Engineering, Faculty of Engineering (v4, p86) : Next:618-181 | Prev:618-171
2. Civil Engineering, Faculty of Engineering (v4, p100) : Next:618-181 | Prev:618-171

618-172 Mathematics 1Q

Credit points: 14.2

Coordinator: Prof. J. H. Rubinstein, Ms. C. Mangelsdorf

Prerequisite: Mathematics 618-171

Contact: 52 hours of lectures (4 a week) and 26 hours of tutorials (2 a week)

Timetable: Semester two

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
• the mathematical formulation of physical problems and their solution using differential equations

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
• skills to apply differential equation techniques to simple problems

Appreciate:

• the role of series in estimation of functions
• the role of linear algebra to find invariants and bring out the underlying geometry in problems
• the similarities and differences between functions of one variable and multivariate functions
• the power of differential equation modelling in advancing an understanding of complex physical processes from a wide variety of real-world phenomena

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, dimensionality, Rⁿ and Cⁿ; inner products; linear transformations, matrix of a linear transformation, change of basis, rank, inverse, solution of linear equations; eigenvectors and eigenvalues, quadrics and conics, rotation matrices, diagonal, real-symmetric and orthogonal matrices. 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, surface areas and volumes of solids of revolution; introduction to double and triple integrals. Systems of differential equations: systems of first order differential equations; linear systems; eigenvalues and eigenvectors; solutions for distinct, repeated and complex eigenvalues; inhomogeneous systems; application to phase plane; equilibrium points and their stability; second order systems, application to systems of mechanical or electrical oscillators; longitudinal and transverse oscillators.

Assessment:

Up to 35 pages of written assignments and up to four hours of end-of-semester written examination (one hour of which will be a written examination on differential equations) and in addition class tests totalling not more than 1.5 hours.

First Year Engineering subject : Next:618-181 | Prev:618-171 | Search | Help
Handbook 1996 : Faculty of Engineering (Volume 4 page 86)

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.