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# Department of Mathematical Sciences

# MATH1561 Single Mathematics A

Please note: This module is not available to Mathematics students.

This module follows on from A-level mathematics, although many topics will be covered afresh. There are three lectures and one tutorial per week. Problems are set to be handed in each week and there is a compulsory examination (Collections) in January. These are all integral parts of the module.

It is important to do the written work conscientiously throughout the year both to prepare yourself for the examination and because there is continuous assessment for written work.

The material consists of important basic ideas and techniques in calculus and linear algebra which have applications in a huge variety of areas of science and mathematics.

## Outline of Course

Aim:

### Term 1

• Elementary Algebra and Basic Functions [Riley Ch. 1] (4 lectures): Simple functions and equations, trigonometric identities, coordinate geometry. Binomial expansion, properties of binomial coefficients. Some particular methods of proof.
• Integration [Riley Ch. 2] (10 lectures): Fundamental theorem of calculus. Natural logarithm; hyperbolic functions. Basic methods of integration including substitution, integration by parts, partial fractions, reduction formulae. Applications of integration.
• Complex Numbers [Riley Ch. 3] (7 lectures): Addition, subtraction, multiplication, division, complex conjugate, modulus, argument, polar form. Argand diagram, de Moivre's theorem, ℯ. Trigonometric and hyperbolic functions. Roots of unity, solutions of simple equations in terms of complex numbers, the fundamental theorem of algebra.
• Limits and Real Analysis [Riley Ch. 2, 4.7] (8 lectures): Real numbers versus rational numbers; limits, continuity, differentiability. Basic methods of differentiation. Utilitarian treatment of the Intermediate Value Theorem, Rolle's Theorem, Mean Value theorem. L'Hopital's rule.

### Terms 2 & 3

• Series and Taylor's theorem [Riley Ch. 4] (10 lectures): Summation of series, convergence of infinite series, absolute and conditional convergence. Taylor polynomials, Taylor's theorem with Lagrange form of the remainder. Convergence of Taylor series. Applications and simple examples of Taylor series.
• Linear equations and matrices [Riley Ch. 8] (16 lectures): Systems of linear equations. Gaussian elimination. Vector spaces, linear operators. Matrix algebra, addition and multiplication, identity matrix and inverses, transpose of a matrix. Determinants and rules for manipulation. Eigenvalues and eigenvectors. Applications to the solution of linear ODEs with constant coefficients.
• Groups [Riley Ch. 28] (7 lectures): Axioms. Simple examples of finite groups. Non-abelian groups. Mapping between groups.

### Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook.