Module Identifier PH26010  
Academic Year 2000/2001  
Co-ordinator Dr Andrew Evans  
Semester Semester 1  
Other staff Dr James Whiteway, Dr Nicholas Mitchell  
Pre-Requisite Core Physics Modules at Level 1  
Course delivery Lecture   18 lectures  
  Seminars / Tutorials   5 seminars/workshops/exercise classes; 6 tutorials  
Assessment Exam   End of semester examinations   70%  
  Course work   Example Sheets Coursework Deadlines (by week of semester): Example Sheets 1 to 10 Weeks 2 to 11   30%  

Module description
A wide variety of problems in Physics can be described in terms of vector operators and differential equations. By the end of the module the student should be able to express common physical systems and relationships using the mathematical language of vectors and differential equations. Topics covered include scalar and vector triple products, description of motion in polar co-ordinates, scalar and vector fields in physics, gradient of a scalar field, divergence and curl of a vector field, vector calculus in cylindrical and spherical polar co-ordinates, eigenvalue problems, general order linear ordinary differential equations, simultaneous differential equations, partial differential equations.

The relevance of the above topics to the physical sciences is illustrated throughout with examples drawn from electrostatics, magnetism, gravitation, mechanics, plasma physics and fluid dynamics.

Learning outcomes
After taking this module students should:

Additional learning activities
Practice at problem-solving is an essential element of Mathematics learning. For this reason, this module combines Lectures with Workshop sessions, where you will get a sheet of questions which you should start attempting during the session itself. A member of staff will be there to help you should you encounter any difficulties with any of the questions. Your marks will constitute the 30% continuous assessment element of the module.

Outline syllabus
Scalar and vector products, highlighting the value of vector notation by application (and simplification) of some physical situations.
2D motion using (r,0) co-ordinate system. Scalar and vector fields tying in with existing knowledge of electrostatics, magnetism, gravitation and pressure along with methods of visualising these fields i.e. contour plots, vector arrow plots.
Gradient of a scalar field. Line integrals. Divergence of a vector field in terms of limiting flux from volume element, and derivation of the divergence theorem.
Applicaitons from electrostatics and fluid dynamics. Curl of a vector field defined in terms of limiting line integral of a loop and Stokes's theorem.
Applications from magnetism and fluid dynamics. Cylindrical and spherical polar co-ordinates. Divergence and curl of vector fields derived from physical definitions and the relevant infinitesimal volume element.

Matrix diagonalization. Eigenvectors and eigenvalues.

Linear O.D.Es with constant coefficients: homogeneous and inhomogeneous.
Linear O.D.Es with non-constant coefficients: Series solutions (the Frobenius Method). Simultaneous linear O.D.Es with constant coefficients: Method of elimination.
Partial differential equations: Method of separation of variables.

Reading Lists
** Recommended Text
Mary Boas, ed.. Mathematical Methods in the Physical Sciences. John Wiley
** Supplementary Text
Stephenson. Mathematical Methods for Science Students.
James. Modern Engineering Mathematics.
James. Advanced Modern Engineering Mathematics.