PX1121 MECHANICS AND MATTER - Cardiff School of Physics and ...

PX1122 MATHEMATICAL METHODS FOR PHYSICISTS I
MODULE INFORMATION:
Autumn semester: 10 credits.
Module organiser: Prof B S Sathyaprakash.
Deputy Module Organiser: Dr E Pascale.
Teaching and feedback methods: Lectures 22 x 1 hr, weekly exercise classes, marked exercises.
Assessment: Examination 60%. Coursework 40%. [Examination duration: 2 hours]
Pre-cursors: None.
Co-requisites: None.
Pre-requisites: None.
AIMS OF THE MODULE:
To provide the basic theoretical techniques required during first-year physics courses.
To introduce post-A-Level students to vectors, matrices, series, functions and graphs, and elementary calculus.
To give students practice in applying mathematical techniques to abstract and physical problems.
LEARNING OUTCOMES:
The student will be able to:
Use vectors to describe positional coordinates and lines and relate them to Cartesian coordinates and show proficiency at
basic vector operations and scalar and vector products and their applications. Application of vector algebra to solve
simple problems in physics.
Demonstrate an understanding of how to perform and exploit basic matrix manipulations.
Show familiarity with and make use of series and expansions of trigonometric, exponential and logarithmic functions.
Show proficiency in plotting graphs of functions and their derivatives and at differentiation, partial differentiation and
integration and their applications.
OUTLINE SYLLABUS:
1. Vectors. Cartesian vectors. Importance in physics. Position. Force. Acceleration. Velocity, etc.
2. Operations on Vectors. Dot and cross product. Angular momentum.
3. Physical interpretation and examples of vectors.
4. Matrices. Definition and interpretation. Addition and multiplication. Matrices operating on vectors, e.g. rotation in the
plane.
5. Inverse of a matrix. Determinants. Use for solving simultaneous equations. Symmetric, anti-symmetric and orthogonal
matrices. Eigenvectors and eigenvalues for symmetric matrices.
6. Functions of a real variable. Exponential, trigonometric and hyperbolic functions. Physical examples. Series expansions.
Binomial and Taylor series. Approximations using truncated series.
7. Differentiation of powers, logarithms, exponential and trigonometric functions. Product, quotient and chain rules. Implicit
and parametric differentiation.
8. Integration of powers, logarithms, exponential and trigonometric functions. Definite and indefinite integrals.
9. Area under a curve. Transforming variables. Integration by parts.
10. Line integrals, multiple integrals, volumes. Curvilinear coordinates, e.g. spherical polar coordinates, introduced through
volume elements.
11. Solution of linear first-order ODEs by separation of variables and by integrating factor.
TRANSFERABLE SKILLS:
Mathematics. Problem solving. Analytical skills.
READING LIST:
Modern Engineering Mathematics, G James (Addison Wesley).
Mathematical Methods for Physics and Engineering, K F Riley, M P Hobson and S J Bence (Cambridge University Press).