*MATH1006 *Mathematical Methods for Physical Scientists 1a

## Module Overview

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 MATH1006 is aimed at physics students. Students taking degrees related to other physical sciences such as chemistry, geology, and oceanography should take MATH1008. 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;
- Algebraically manipulate 3 by 3 matrices, and solve eigenvalue problems in 2 dimensions;
- 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

• calculate the scalar and vector product of two vectors; • algebraically manipulate 3 by 3 matrices, and solve eigenvalue problems in 2 dimensions; • solve simple polynomial equations, including complex solutions • 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; • understand the nature of simple complex valued functions; • differentiate functions and manipulate them when changing variables; • integrate various functions of one variable. • Basic vector algebra, cross and dot product, geometrical and physical applications. Matrices and determinants, inverse of a matrix, using matrices to solve simultaneous equations. Eigenvalue problems. Solving quadratic equations, factorising higher order polynomials. Complex numbers, powers of i, Argand diagrams, modulus and argument of a complex number, complex conjugates. The algebra of complex numbers: addition, subtraction, multiplication and division in both Cartesian and polar form. Specifying a function, its domain and range. Composition of functions. Graphs of functions. One-toone 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. Exponential and natural logarithm functions, power functions, hyperbolic functions, inverse hyperbolic functions and their derivatives. Derivatives of vectors L'Hôpital's Rule, Taylor series expansions and remainder terms. Complex exponentials and trigonometric functions. De Moivre's Theorem, calculating powers and roots, and solving equations. Integration, the Fundamental Theorem of Calculus, indefinite integrals, methods of integration, partial fractions, integration by parts.

### Learning and Teaching

#### Teaching and learning methods

Lectures, small group tutorials, private study. The method of delivery in lectures will be “chalk and talk”, however the students will be provided with printed skeletal notes, highlighting all the key results and saving them from excessive note taking. Hard copies of the skeletal notes and all the assignments will be provided, and the material will also be provided in the module Blackboard site.

Type | Hours |
---|---|

Independent Study | 150 |

Total study time | 150 |

#### Resources & Reading list

ADAMS R A. Calculus - a Complete Course.

### Assessment

#### Summative

Method | Percentage contribution |
---|---|

Coursework | 10% |

Exam | 80% |

Problem Sheets | 10% |

#### Referral

Method | Percentage 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.