MPhys Physics with Astrophysics

Mathematics 2

To acquire the skills in vector calculus needed to understand Electromagnetism, Fluid and Quantum Mechanics. To acquire an introductory understanding of Fourier Series and their use in physics.

On completion successful students will be able to:

1. Explain the concepts of scalar and vector fields.

2. Describe the properties of div, grad and curl and be able to calculate the divergence and curl of vector fields in various coordinate systems.

3. Calculate surface and volume integrals in various coordinate systems.

4. Calculate flux integrals and relate them to the divergence and the Divergence Theorem.

5. Calculate line integrals and relate them to the curl and to Stokes' Theorem.

6. Apply the methods of vector calculus to physical problems.

7. Calculate the Fourier series associated with simple functions and apply them to selected physical problems.

1.  Differentiation and integration with multiple variables                                                                         (6 lectures)

Partial and total derivatives. Taylor’s theorem for multivariable functions. Multiple integration over areas and volume; volumes, masses and moments of inertia. Use of limits in integrals. Methods of evaluation of multiple integrals. Cylindrical and spherical polar coordinates.  Jacobian Determinant.

2.  Vector operators: div, grad and curl                                                                                                          (6 lectures)

Scalar and vector fields. Definition and uses of the gradient operator. The method of Lagrange multipliers. Definitions of divergence and curl. Combinations of div, grad and curl; theorems. The Laplacian. Vector operators in cylindrical and spherical polar co-ordinates.

3.  The Divergence Theorem, Stokes Theorem, conservative forces                                                         (7 lectures)

Line integrals of scalar and vector fields. Surface integrals and flux of vector fields. Integral expression for divergence. Divergence theorem and its uses. Conservation laws; continuity equation. Integral expression for curl. Stokes' theorem and its uses. Definition of conservative field. Relation to potentials.

4. Introduction to Fourier Series                                                                                                                   (3 lectures)

Rationale for using Fourier series. The Dirichlet conditions. Orthogonality of functions. The Fourier coefficients, symmetry considerations. Examples of Fourier series. Complex representation of Fourier Series.

* Other

10% Weekly online 

10% Tutorial Work/attendance 

Feedback will be offered by tutors on students’ written solutions to weekly examples sheets, and model answers will be issued. Interactive feedback will be offered during the Workshop sessions.

Martin, B. R. and Shaw, G. Mathematics for Physicists (Manchester Physics Series, Wiley). 

Rile, K.F., Hobson, M.P. and Bence, S.J. Mathematical Methods for Physics and Engineering

Schey, H. M. Div. Grad, Curl and All that, 2nd ed. (Norton)

M. Boas, Mathematical Methods in the Physical Sciences (3rd Edition, Wiley)