# Links for NST IB Physics A/B `Mathematical Methods' 2022

16 lectures, Michaelmas term, F and M, 11:00am in Cormack Room, University Centre.

### 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

Here are corrections for the printed handouts.

• Handout I, p8: the punctuation of the last equation is mangled, move the "." to the end.
• Handout I, p10, Section 2.1: in the description of cylindrical polar coordinates, the angle should be $\phi$ not $\theta$, as is used subsequently (e.g. p11).
• Handout III, p6 and p7: in equation 6.40 there is an arbitrary constant $G$, which should be included in equation 6.41, and in the text before equation 6.45 (as this constant it has been combined with the other constants into $C_{lmn}$).
• Handout III, p23: just before 8.8 it should say consider the $\rho$ variation.
• Handout III, p29: after 8.42, the text should read "… $h=R/a$, and take out …"
• Handout III, p29: RHS of 8.43 should have extra parentheses to read "… $(R/a)^l$ …"
• Handout IV, p5: the formatting on the lines in the first table on this page could have been clearer (for a transpose conjugate or Hermition conjugate matrix), to read:  $\cdots$ $\ \ \$ $A^{*T}\ \ \$ ${\rm transpose\ conjugate\ \ \ }$ ${\rm complex\ conjugate}$ ${\rm or}$ ${\rm or}$ ${\rm each\ element}$ $A^\dagger$ ${\rm Hermitian\ conjugate}$ ${\rm then\ transpose}$ $\cdots$

Here are .pdfs of additional material shown in the lectures.
• 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, p16/17: algebra for RHS of equation 5.38 to RHS of equation 5.39.
• 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 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 give 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.
• Zernike polynomials: here is a visualisation of Zernike polynomials, which are orthogonal over a unit circle. These are used to characterise aberrations for optical instruments (hence the names).
• Waves (Bessel functions): webpage vitalising 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’.)
• 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 TiS. Also you can purchase a copy for £1.50 from Mr Richard King, in the Part IA practical laboratory.

(last changed 2022 November) 