620-172 Mathematics 1Q

Note

This subject is not available to students enrolled in any Bachelor of Science course. Combined Science/Engineering students are required to take Science Mathematics.

Credit Points

Coordinator

Prerequisites

620-171 (1997 Handbook 618-171).

Semester

Contact

Subject Description

This subject gives a solid grounding in key areas of mathematics needed in modern science and technology, and is the gateway to all mathematical developments since 1900. Little of the material here has been seen at school, and the level of understanding required represents a distinct advance on the minimum necessary in previous subjects.

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 C^n; 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, four hours of end-of-semester written examination (one hour of which will be a written examination on differential equations) and class tests totalling not more than 1.5 hours.