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Department of Mathematical Sciences

# MATH1571 Single Mathematics B

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.

In the first term we will discuss vector algebra and some applications to mechanics and geometry, ordinary differential equations - their classification and solutions, and Fourier analysis - the representation of functions as linear superpositions of sines and cosines.

In the second and third terms we cover functions of several variables, partial differential equations, and probability. The ideas of differentiation and integration extended to functions of two or more variables give rise to partial derivatives and multiple integrals. A partial differential equation expresses a relationship involving a function of two or more variables and some of its partial derivatives. Wave motion is one of the many phenomena described by partial differential equations; an example is vibration of a stretched string, such as a guitar string. The final part of the module provides an introduction to probability.

## Outline of Course

Aim:

### Term 1

• Vectors [RHB chapter 7] (9 lectures): Scalars and vectors. Bases and components. i, j, k notation. Vector algebra. Multiplication of vectors: scalar and vector products and their geometrical meaning, length and orthogonality. Triple products. Applications: equations of lines and planes, distances. Derivatives with respect to scalars: velocity, acceleration forces, moments, angular velocity. Two-dimensional polar coordinates, spherical and cylindrical polar coordinates.
• Ordinary Differential Equations [RHB chapter 14] (12 lectures): General properties. First-order first-degree equations: separable, homogeneous, linear, Bernoulli’s equations. First-order higher-degree equations. Second-order linear equations with constant coefficients. Applications to particle dynamics, using Newton’s Laws of Motion.
• Fourier Analysis [RHB chapter 12] (8 lectures): Periodic functions, orthogonality of trigonometric functions. Dirichlet conditions, Fourier representation and coefficients. Odd and even functions. Complex form. Parseval’s theorem.

### Terms 2 & 3

• Partial differentiation [RHB chapter 5] (9 lectures): Functions of several variables, graphs. Partial derivatives, differential, exact & inexact differentials. Chain rule, change of variables. Solutions of simple partial differential equations, d'Alembert's solution of the wave equation. Taylor expansions, critical points.
• Complex Analysis[Riley section 24] (4 lectures): Differentiability of complex functions; Cauchy-Riemann equations. Power series in a complex variable. Cauchy's Theorem.
• Multiple integration [RHB chapter 6] (7 lectures): Double integrals, in Cartesian and polar coordinates. Triple integrals and integration in cylindrical and spherical polars. Applications. Change of variables in multiple integrals, Jacobians.
• Vector Calculus [RHB chapter 10] (3 lectures): Differentiation and integration of vectors. Vector fields. Grad operator.
• Probability [RHB chapter 30] (9 lectures): Sample space, probability axioms, conditional probability, random variables, independence, probability distributions (binomial and normal distributions), expectation and variance. Application to experimental errors.

### Prerequisites

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