# MATH29661 (Spring 2017)

Undergraduate tutorial sesssions, *The University of Manchester, School of Mechanical, Aerospace and Civil Engineering*, 2017

The course unit aims to provide a second year course in mathematics and statistics to students in the School of School of Mechanical, Aerospace and Civil Engineering

# Syllabus

Multiple and Line Integrals. Construction and evaluation of double integrals including changing the order of the integrations. Change of variable including the Jacobian. Introduction to triple integrals. Further Line integrals.

Vector Calculus. Scalar and Vector fields. Gradient, divergence and curl. Laplacian. Identities. Line and Surface integrals involving vectors. Vector integral theorems.

Laplace Transforms: Definition. Transforms and Inverse Transforms of Simple Functions. Transforms of derivatives and integrals. Convolution, Solution of Ordinary Differential Equations using Laplace Transforms.

Numerical Methods. Interpolation and Least Squares approximation. Further numerical integration, integration rules, numerical integration where interval or function becomes infinite. Solution of systems of non-linear algebraic equations.

General notion of a random variable, including its definition and the range space. Discrete random variables – definition and explanation; probability mass function (pmf); discrete Uniform distribution; Binomial distribution; mean and variance of discrete random variables. Continuous random variables – definition and explanation; probability density function (pdf); Uniform distribution; Exponential distribution; Normal distribution; mean and variance of continuous random variables. Linear transformations of random variables. ie. Y=aX+b. Mean and variance of Y. The cumulative distribution function for discrete and continuous random variables; calculating Normal probabilities (standardising), characteristic load and strength. Normal approximation to the Binomial distribution. Application to quality control.