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Module Title

Mathematics for Physicists 1 B

School

Physics and Astronomy

Department

Physics & Astronomy

Module Code

03 19753

Module Lead

Dr R A Smith, Mr J Kronjaeger

Level

Certificate Level

Credits

30

Semester

Full Term

Pre-requisites

Co-requisites

Restrictions

None

Exclusions

Description

The laws of physics are written in mathematical form, and it is clear that we will need to understand a certain amount of mathematics if we are to solve any physical problems. To see what is required, let us consider Newton¿s second law of mechanics: force equals mass times acceleration. Force and acceleration are vector quantities, having magnitude and direction, whereas mass is a scalar quantity, having only magnitude. It follows that we will have to recognise and manipulate both scalar and vector quantities. Acceleration is the rate of change, or time derivative, of velocity; it follows that we will need to understand differentiation, and its inverse process of integration. If we next consider electromagnetism, we see that electric and magnetic fields exist at every point of three dimensional space, and every instant of time; we must therefore understand functions of more than one variable, and in fact extend the ideas of differentiation and integration to such functions. Finally we may notice that all the major laws of physics involve derivatives, and thus are differential equations; we will need to understand such equations, and how to solve them in various circumstances. This two semester course develops these mathematical techniques needed by physics modules in the first and subsequent years. The sequence of topics is carefully chosen to support the physics modules in the first and second semesters. Where possible, mathematical ideas are linked to physics topics, and there is a strong emphasis on problem solving. The topics covered are:Semester 1: Scalars and vectors; differentiation; complex numbers Semester 2: Functions of several variables; partial differentiation; differential equations; single and multiple integrals.

Learning Outcomes

By the end of the module the student should be able to:

understand the idea of vector and scalar quantities;

perform vector algebra including addition, scalar and vector products up to the level of scalar and vector triple products;

understand and use vector equations for lines and planes;

understand the idea and definition of a derivative;

differentiate using standard rules for addition, product, quotient, function of a function and inverse function;

use differentiation to find maxima, minima and points of inflexion;

curve plotting in cartesian and polar coordinates;

expand functions as Taylor series;

understand the origin of complex numbers via polynomial equations;

perform addition, multiplication and division with complex numbers;

understand and use Euler¿s and de Moivre¿s theorem;

define and use modulus, argument and complex conjugate;

define and use functions of several variables and partial derivatives;

extend the chain rule, theory of maxima and minima and Taylor series to functions of more than one variable;

classify differential equations by degree, order, homogeneity, linearity and ordinary vs partial equations;

solve first order ordinary differential equations (o.d.e.) of separable, exact, homogeneous and linear type;

understand importance and properties of linear differential equations;

solve linear o.d.e. of any degree with constant coefficients;

understand the definition of integration as area under a curve;

integrate functions of a single variable using standard methods: inspection, integration by parts, substitution, trigonometric identities;

understand the definition of integration as area under a curve;

integrate functions of a single variable using standard methods: inspection, integration by parts, substitution, trigonometric identities;

understand the idea of and evaluate multiple and repeated integrals;

change order of integration and change variable in multiple integrals;

use single or multiple integrals to evaluate area, volume, mass, centre of mass and moment of inertia;

show fluency in all the above mathematical techniques so that mathematics is not an obstacle in solving physical problems.

Assessment

19753-01 : Examination : Exam (Centrally Timetabled) - Written Unseen (60%)
19753-02 : Class Test : Coursework (20%)
19753-03 : Examples Sheets (sem1) : Coursework (20%)

Assessment Methods & Exceptions

One 3hr written examination in the summer term (60%); 20 assessed problems (20%); 4 class tests (20%)