The University of Southampton
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Physics and Astronomy

MATH1006 Mathematical Methods for Physical Scientists 1a

Module Overview

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 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 & Objectives

Aims

Aim

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-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. 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 & Teaching

Learning & teaching 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.

ActivityDescriptionHours
LectureLectures, small group tutorials, private study. The method of delivery in lectures will be “chalk and talk”, however the students will be provided with 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 also mounted on the module Webpage.
TutorialLectures, small group tutorials, private study. The method of delivery in lectures will be “chalk and talk”, however the students will be provided with 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 also mounted on the module Webpage.

Assessment

Assessment methods

MethodHoursPercentage contribution
Closed book examination-80%
For coursework consisting of marked weekly problem sheets; Referral assessment: written examination. A module Formula Sheet will be provided and a copy of this may be used in the examination-10%
Self-marked problem sheets-10%
Exam hours80%

Referral Method: By examination

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