618-172 Mathematics 1Q

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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.

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