618-200 Mathematics 2

Note:

1. Students may not gain credit for more than one of 618-200, 618-211, 618-102 (1995 Handbook), 618-112, 618-122.

2. The content of 618-200 will be revised for 1998, reflecting changes in 618-141 and 618-142 in 1997.

Credit Points:

Coordinator:

Prerequisite/s:

Timetable:

Contact:

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 to find invariants and bring 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, 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.

Assessment: