421-204 Mathematics for Engineers Unit 2.1B (Basic)

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Objectives:

On completion of this course students should have:

• a fundamental knowledge of both series and matrix methods. Series are essential to the understanding of many phenomena over small distances and for small times. Matrix methods are important in many areas of engineering especially in the development of numerical procedures;

• a background in multivariable calculus which forms the foundations of the study of continuous phenomena in more than one dimension, for example as commonly occur in heat transfer.

Content:

Part 1 Sequences, Series and Matrices: Convergence and divergence of sequences and series, tests for convergence. Taylor's theorem and series representation of elementary functions. Eigenvectors and eigenvalues, linear independence, factorisation of matrices, diagonal, real-symmetric, Hermitian, unitary and orthogonal matrices, quadric surfaces, quadratic forms, definiteness of matrices.

Part 2 Multivariable Calculus: Functions of several variables, level curves, heights. Partial derivatives, total derivative, gradient vector, directional derivatives and applications. Chain rules. Coordinate transformations, Jacobi matrix and determinant. Maxima and minima of functions of several variables with constraints. Lagrange multipliers. Area and volume integrals, iterated integrals, Cartesian and polar coordinates in two and three dimensions.

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