$$ \def\ab{\boldsymbol{a}} \def\bb{\boldsymbol{b}} \def\cb{\boldsymbol{c}} \def\db{\boldsymbol{d}} \def\eb{\boldsymbol{e}} \def\fb{\boldsymbol{f}} \def\gb{\boldsymbol{g}} \def\hb{\boldsymbol{h}} \def\kb{\boldsymbol{k}} \def\nb{\boldsymbol{n}} \def\tb{\boldsymbol{t}} \def\ub{\boldsymbol{u}} \def\vb{\boldsymbol{v}} \def\xb{\boldsymbol{x}} \def\yb{\boldsymbol{y}} \def\Ab{\boldsymbol{A}} \def\Bb{\boldsymbol{B}} \def\Cb{\boldsymbol{C}} \def\Eb{\boldsymbol{E}} \def\Fb{\boldsymbol{F}} \def\Jb{\boldsymbol{J}} \def\Lb{\boldsymbol{L}} \def\Rb{\boldsymbol{R}} \def\Ub{\boldsymbol{U}} \def\xib{\boldsymbol{\xi}} \def\evx{\boldsymbol{e}_x} \def\evy{\boldsymbol{e}_y} \def\evz{\boldsymbol{e}_z} \def\evr{\boldsymbol{e}_r} \def\evt{\boldsymbol{e}_\theta} \def\evp{\boldsymbol{e}_r} \def\evf{\boldsymbol{e}_\phi} \def\evb{\boldsymbol{e}_\parallel} \def\omb{\boldsymbol{\omega}} \def\dA{\;d\Ab} \def\dS{\;d\boldsymbol{S}} \def\dV{\;dV} \def\dl{\mathrm{d}\boldsymbol{l}} \def\rmd{\mathrm{d}} \def\bfzero{\boldsymbol{0}} \def\Rey{\mathrm{Re}} \def\Real{\mathbb{R}} \def\grad{\boldsymbol\nabla} \newcommand{\dds}[2]{\frac{d{#1}}{d{#2}}} \newcommand{\ddy}[2]{\frac{\partial{#1}}{\partial{#2}}} \newcommand{\pder}[3]{\frac{\partial^{#3}{#1}}{\partial{#2}^{#3}}} \newcommand{\deriv}[3]{\frac{d^{#3}{#1}}{d{#2}^{#3}}} \newcommand{\ddt}[1]{\frac{d{#1}}{dt}} \newcommand{\DDt}[1]{\frac{\mathrm{D}{#1}}{\mathrm{D}t}} \newcommand{\been}{\begin{enumerate}} \newcommand{\enen}{\end{enumerate}}\newcommand{\beit}{\begin{itemize}} \newcommand{\enit}{\end{itemize}} \newcommand{\nibf}[1]{\noindent{\bf#1}} \def\bra{\langle} \def\ket{\rangle} \renewcommand{\S}{{\cal S}} \newcommand{\wo}{w_0} \newcommand{\wid}{\hat{w}} \newcommand{\taus}{\tau_*} \newcommand{\woc}{\wo^{(c)}} \newcommand{\dl}{\mbox{$\Delta L$}} \newcommand{\upd}{\mathrm{d}} \newcommand{\dL}{\mbox{$\Delta L$}} \newcommand{\rs}{\rho_s} $$
Preamble
These are the lecture notes for MATH 2811 Mathematical Methods II for the second half of Michael Term 2025/26. I will generally follow these notes in the lectures, although not word for word.
If you do spot any significant discrepancies, or typos in the notes, then please let me know!
Many aspects of the notes draw on those inherited from previous lecturers Anne Taormina as well as Andrew Krause, Andreas Muench and Derek Moulton. It borrows heavily form the style of Anthony Yeates in term 1.
The nature of the topic is less compartmentalised than in the previous half of the course, and the notes largely build up to an overarching mathematical principle; the spectral or eigenfunction representation of solutions to partial differential equations. There are few theorems and a lot of he material concerns putting some of the techniques of calculus 1 under a fine tooth comb, and asking how they generalise (the answer significantly). Thus the style is slightly different so I have a slightly expanded different color scheme for my side boxes.
So we have:
Non-examinable material to give extra context.
Here be dragons…..
The key new box type. Some of the material involves quite lengthy intricate explanations. These boxes are used to summarise the kep points you need to take for what follows for the rest of the course.
The notes are formatted with Quarto (https://quarto.org/docs/books).