# Unit information: Maths with Numerical Modelling for Physics in 2020/21

Unit name Maths with Numerical Modelling for Physics PHYS10008 20 C/4 Teaching Block 2 (weeks 13 - 24) Professor. Annett Not open Normally A-level Physics and A-level Mathematics or equivalent. PHYS10006 Core Physics I, PHYS10005 Core Physics II, MATH11004 Mathematics 1A20 School of Physics Faculty of Science

## Description

This unit will provide practice and training in the mathematics needed to complete the first year Physics courses and lay the foundations for subsequent years.

Topics covered include:

• The equation of heat conduction, and its solution by half-range Fourier series
• Partial differentiation, the gradient vector and its physical meaning
• Contours; tangents and normals to curves
• Change of variables and the chain rule
• Maxima and minima; stability of equilibrium
• Parametric curves, line integrals and work done by a force; conservative fields
• Exact differentials
• Double integrals, including change of variables and polar coordinates; application to moments of inertia
• Green's Theorem relating line integrals to double integrals; application to magnetic field generated by a current
• Matrix algebra, matrices as transformations of vectors, rotation and reflection matrices
• Determinants. Inverse matrix. Eigenvalues of 2 x 2 and 3 x 3 matrices, and application to vibrations.

Further, the course will introduce basic computer programming in Python to permit students to explore the mathematical concepts above through:

• evaluation of functions and series
• plotting functions and data in 2-D and 3-D
• simple matrix algebra solutions to sets of linear equations.

Aims:

• To motivate students to learn mathematics, by showing it in action in physics
• to develop students' mathematical skill and introduce the mathematical tools needed for first-year Physics
• to introduce basic programming skills useful for illustrating mathematical and physical principles, which will be built upon in computing courses in later years.

## Intended learning outcomes

After completing this unit students should:

• be able to solve problems using partial differentiation, line integrals, double integrals, Fourier series, matrix algebra, and calculation of eigenvalues and eigenvectors of simple 2 x 2 and 3 x 3 matrices
• have an appreciation of the physical meaning and application of: the gradient vector, line integrals and conservative fields, Fourier series, and eigenvalues
• be able to write and test basic scientific programs using Python
• be able to visualise simple mathematical functions by generating plots in 2-D and 3-D
• be able to demonstrate principles of linear algebra using numerical libraries in Python.

## Teaching details

The unit will be taught through a combination of

• asynchronous online materials, including narrated presentations and worked examples
• synchronous group problems classes, workshops, tutorials and/or office hours
• asynchronous directed individual formative exercises and other exercises

## Assessment Details

Weekly mathematics problems are both formative (through discussion in tutorials and written feedback) and summative.

The computing will be assessed formatively through regular online assessment of individual and pair-based work. The computing coursework and final extended exercise will be assessed summatively.

The final assessment mark for the unit is made up of:

• Mathematics weekly set written/e-assessment problems (10%)
• Computing coursework (30%)
• Mathematics examination (40%)
• Computing extended exercise (20%).