Links for NST IB Physics A/B `Mathematical Methods' 2022
Handouts |
Corrections |
Additions |
Resources |
Books/Formula Handbook
16 lectures, Michaelmas term, F and M, 11:00am in Cormack Room,
University
Centre.
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.
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.
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.
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.
Dave Green –
(last changed 2022 November)
