Links for NST IB Physics A/B ‘Mathematical Methods’ 2023

Handouts | Corrections | Additions | Resources | Books/Formula Handbook

16 lectures, Michaelmas term, Friday and Monday, 11:00am.

Note: these lectures are in different rooms on different days.

• The Friday lectures are in Titan Teaching Room 1 (see map);
• The Monday lectures are in Titan Teaching Room 2 (see map).

1) Handouts

The handouts and problem sheets for this course are available from the course page of the Cavendish Teaching Information System (TiS) website, for registered users.

2) Corrections

• Handout I, p19: in the Example, it should say that the region from O to B is when $y>0$.
• Handout II, p8: after equation 4.46 add ‘So for $t=0$ then’, and change equation 4.47 to read $B_m(0)=C$.
• Handout II, p22: for equation 5.90, the RHS should say the first result is for ‘if $0 < x < x'$’ and the second is for ‘if $x' < x < L$’ (as is given in 5.96).
• Handout III, p9: in the table, $q(x)$ for the Legendre equation should be $0$ and for the associated Legendre equation should be $\frac{-m^2}{1-x^2}$.

3) Additions

• Handout I, p15: alternative ways of writing curl in cylindrical polar coordinates.
• Handout I, p16/17: graph showing the stationary points for distance from the origin with the constraint $y=1-x^2$.
• Handout II, p2/3: plot to illustrate why for the Fourier series of a square wave the amplitudes of even values $n$ are zero (in this case $n=2$).
• Handout II, p18/19: Fourier transform of cosine (and sine), to show symmetries between $f(t)$ and $F(\omega)$.
• Handout II, p20/21, Section 5.8: Fourier transforms of multiple $\delta$ functions.
• Handout II, p22: algebra required to go from equation 5.93 and 5.94 to 5.95.
• Handout III, p11/12: showing in a bit more detail how to derive equations 7.16, 7.19, 7.20 and 7.21.
• Handout III, p26: alternate ways of writing spherical harmonics, to obtain $s, p, d$ orbitals as used in chemistry.

4) Resources on the Web

As the course progresses I will provide here links to resources on the Web that relate to various topics discussed in the lectures.

• Convolution: a webpage that illustrates the convolution of two functions with an animation, which has a choice of various functions. Note, you can interactively change the width of the functions before starting the convolution animation by click on the dot in the plot of a function, and drag to change.
• Waves (Bessel functions): webpage visualising of waves on a circular membrane, which is Q24. (This is from this website, which has other useful visualising of various maths/physics, including the links below for ‘Normal Modes’.)
• Zernike polynomials: here is a visualisation of Zernike polynomials, which are orthogonal over a unit circle. The left plot of each is the phase variation over the circle, e.g. a lens, and the right plot shows the distortion in the image from the lens due to the phase variation. These are used to characterise aberrations for optical instruments (hence the names).
• Spherical Harmonics: visualisations of spherical harmonics (here used as the basis set to describe the gravitational field/geoid (i.e. equipotential surface) of the Earth).
• Normal Modes: webpages that visualise normal modes for 1-D motion of multiple masses coupled by springs, and sideways oscillations of masses on a string.

5) Books/Formula Handbook

There are many books that cover the material in this course, including the following.

• ‘Mathematical Methods in the Physical Sciences’, by Boas M L (3rd edition, Wiley 2006).
• ‘Mathematical Methods for Physics and Engineering’, by Riley, Hobson & Bence (3rd edition, CUP 2006). (This covers many more advanced topics also. As this is published by CUP you can read this online. Also available is the earlier 2nd edition, where chapters are available in .pdf format.)
• The ‘Mathematical Formula Handbook’ – which is provided to you in NST Physics examinations – is available here (version 2.5), or on the Cavendish website. Also you can purchase a copy for £1.50 from Mr Richard King, in the Part IA practical laboratory.

Dave Green –

(last changed 2023 November)