The University of Southampton
Courses

# MATH1008 Introduction to MathematicalMethods

## Module Overview

This module is suitable for students with A level Mathematic (grade B or higher). – Students with AS level Mathematics are required to take MATH1004 instead. The aim of the module is to provide students with the necessary skills and confidence to apply a range of mathematical methods to problems in the physical sciences. Both MATH1006 and MATH1008 cover essentially the same topics in calculus that are of relevance to applications in the physical sciences but MATH1008 is aimed at students taking degrees in chemistry, geology and oceanography. Physics students should take MATH1006. The aim of the module is to provide students with the necessary skills and confidence to apply a range of mathematical methods to problems in physics. The module begins by looking at vectors in 2 and 3 dimensions, introducing the dot and cross products, and discussing some simple applications. This is followed by a section on matrices, determinants, and eigenvalue problems. The course then reviews polynomial equations and introduces complex numbers. After this, some basic abstract concepts related to functions and their inverses are discussed. The main part of the unit covers the basics of calculus, starting with limits, and going on to look at derivatives and Taylor series. The concept of integration is then defined, followed by an exploration (by means of examples) of various methods of integration.

### Aims and Objectives

#### Module Aims

To provide students with the necessary skills and confidence to apply a range of mathematical methods to problems in the physical sciences.

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

• Calculate the scalar and vector product of two vectors;
• Solve simple polynomial equations
• Sketch and manipulate exponential, trigonometric and hyperbolic functions
• Differentiate functions of one variable and use this to classify critical points
• Understand the concept of a limit and be able to determine its value if it exists
• Construct a Taylor series of a function and understand its relevance to local behaviour;
• Differentiate functions of several variables and manipulate then when changing variables
• Integrate various simple functions of one variable;

### Syllabus

Basic vector algebra, cross and dot product, geometrical and physical applications. Matrices and determinants, inverse of a matrix, using matrices to solve simultaneous equations. Solving quadratic equations, factorising higher order polynomials. Specifying a function, its domain and range. Composition of functions. Graphs of functions. One-to-one functions, inverse functions and their graphs. Even and odd functions, periodic functions, trigonometric functions, inverse trigonometric functions. Informal definition of a limit, rules for evaluating limits, infinite limits. Rules for differentiation, higher derivatives, critical points and applications to graph sketching. Method of least squares. Exponential and natural logarithm functions, power functions, hyperbolic functions, inverse hyperbolic functions and their derivatives. L'Hôpital's rule Rule, Taylor series expansions. Integration, the Fundamental Theorem of Calculus, indefinite integrals, methods of integration, partial fractions, integration by parts. Functions of two and more variables, partial derivatives of first and higher order. Chain rules. Transformations of second derivatives. Surfaces and critical points. Derivatives of vectors.

### Learning and Teaching

TypeHours
Teaching48
Independent Study102
Total study time150

ADAMS R A. Calculus - a Complete Course.

### Assessment

#### Summative

MethodPercentage contribution
Assignments and problem sheets 10%
Coursework 10%
Exam 80%

#### Referral

MethodPercentage contribution
Exam 100%

#### Repeat Information

Repeat type: Internal & External

### Costs

#### Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

##### Books and Stationery equipment

Course texts are provided by the library and there are no additional compulsory costs associated with the module.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at www.calendar.soton.ac.uk.