618-182 Mathematics 1S

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Objectives:

On completion of this subject students should:

Comprehend:

• some of the nature of the different types of numbers they use;

• the intuitive nature of limits as used in continuity, differentiation and integration;

• the notion of integral as area;

• 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 manipulate complex numbers and to use them to solve problems;

• skills in approximation and estimation;

• 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 proof and logical reasoning in mathematics;

• the use of complex numbers;

• the role of limits in both the differential and integral calculus;

• the practical uses of calculus;

• the power of differential equation modelling in advancing an understanding of complex physical processes from a wide variety of real world phenomena.

Content:

Foundations: sets, integers, mathematical induction; real numbers; complex numbers, polar form, de Moivre's theorem, complex exponential. Calculus: functions of one real variable (including limits and continuity), Mean Value Theorem and applications, Newton's method for root-finding, approximate integration, Taylor polynomials. 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; introduction to double and triple integrals and applications. 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 oscillations.

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