Partial Differential Equations/Stylistic guidelines

Mostly taken from Prof. Arieh Iserles' course 'How to write mathematics':

Language

• Include many explanations and examples while being as brief as possible.
• Include occasional jokes (if you are funny, please include some, because the main author is not funny).
• This wikibook is to be written in BRITISH english.

Proofs

• Only leave trivial things to the reader.
• Put complicated and very technical results into the appendix.
• Put the parts of proofs which are 'pure calculation' into lemmata such that the proof of a theorem also serves as the starting point for developing an internal proof synopsis.

Theorems

• Always mention the weaknesses of theorems.

Structure

• Let the structure follow the intuitive comprehension process of the reader.
• Make the structure conform to every possible leaning structure (e.g. learning the theorems and definitions first, learning linear etc.).
• Use roughly equal sizes for same-level sections.
• Keep lowest level sections short.
• Include Illustrations by examples, tables and figures.
• Introduce new concepts just before they are needed.
• Put important theorems in a textbox.

Links outward

• Include as many links to other Wikimedia pages as possible
• Do not link to unofficial/commercial pages or unethical journals

Figures

• Only include figures if they make a point; they shouldn't be included if they are only ornamental.
• Make the figures easy to understand.
• Link the figures to the text.

Notation

• Avoid too many subscripts, tildes, multiple indices, hats etc.
• Recall definitions if they have not been used a long time and are now to be used again.
• Don't overload notation; variables should have only one meaning.
• Don't use two different notations for the same thing.
• Use the following notation conventions throughout the book (note that we distinguish between boldface, upper case, lower case, ...) (the priority is given by the order):
  • letter for generic element of a set: $x$
  • letters for vectors of generic vector space (for a generic vector in ${ℝ}^d$ please use $x$ and $y$, see below at the notation for the spatial variable): $𝐮$, $𝐯$, $𝐰$
  • letters for vector constants: $𝐛$, $𝐜$
  • letters for solutions of pde's: $u$, $v$, $w$
  • letter for a smooth function $B\to ℝ$ in linear partial differential equations: $a$
  • letters for constants which are elements of a field: $b,c$
  • letter for element of $[0,1]$: $\lambda$
  • letter for spatial dimension: $d$
  • letters for bump functions: $\phi$, $\vartheta$
  • letters for Schwartz functions: $\psi$, $\theta$
  • letter for sets not assumed to be open or closed: $S$
  • letters for open sets: $O$, $U$
  • letter for closed sets: $A$
  • letter for domains: $\mathrm{\Omega }$
  • letter for compact sets: $C$
  • letter for convex sets: $Q$
  • letter for generic set: $X$
  • letter for metric space: $M$
  • letter for generic vector space: $V$
  • letter for topology: $\tau$
  • letter for generic topological space: $𝒳$
  • letter for generic topological vector space: $𝒱$
  • letter for generic function: $f$
  • letter for function of inhomogenous problems: $f$ (since this is the convention in many sources)
  • letter for diffeomorphism: $\psi$
  • letter for outward normal vector: $\nu$
  • letter for hessian matrix of $f\in {𝒞}^2(O)$: $H_f$
  • letters for initial/boundary conditions: $g$, $h$
  • letter for auxiliary function (and its variable): $\mu (\xi )$
  • letter for curve (and its variable): $\gamma (\rho )$
  • letters for vector fields: $𝐕$, $𝐖$
  • letters for multiindices: $\alpha$, $\beta$, $\varrho$, $\varsigma$
    • Priority: Generic multiindex in that order, summation index in reversed order
  • letters for time and space: $t$, $x$ (i know the space variable is already used for the elements of sets but that is a wide-spread convention)
  • secondary letters for time and space and arguments of the Fourier transform: $s$, $y$
  • tertiary letter for space: $z$ (unfortunately, but there is no other suitable candidate)
  • letter for radius: $R$
  • notation for area and volume of $d$-dimensional sphere with radius $R$: $A_d(R)$, $V_d(R)$
  • letter for generic fundamental solution: $F$
  • notation for Green's kernels:
    • Generic green's kernel: $K$
    • Green's function: $G$
    • Poisson's equation: $P$
    • Heat equation: $E$
    • Helmholtz' equation: $Z$
  • letters for generic natural number and summation indices: $n,k,j$
    • Priority: For summation $j,k,n$, for generic natural number $n,k,j$
  • letters for sequence indices: $l,m$
  • letters for natural numbers above which something holds: $N,J,M$
  • notation for $d$-dimensional multiindex consisting only of $l$s: $\varrho (d,l)$
  • imaginary unit: $i$
  • Euler's constant: $e$
  • letter for linear functions: $T$
  • fundamental lagrange polynomial: ${\ell }_{k,x_1,\dots ,x_n}$
  • Interpolating polynomial: $L_{f,x_1,\dots ,x_n}$
  • letter for linear and continuous functions: $ℒ$
  • letter for members of a dual space: $𝒯$ (for regular (tempered) distributions generated by $f$: ${𝒯}_f$)
  • letter for the Gaussian function: $\varphi$
  • sets defined by conditions: $\{x\in \text{a set}|x\text{ satisfies a condition}\}$
  • element in index set: $\upsilon \in \mathrm{{\rm Y}}$
  • letter for set of continuous functions: $𝒬$
• In arguments of solutions of time-dependent partial differential equations, write the time variable first and then the space variable.
• For sums, write down the complete substack, except when dealing with natural numbers.
  • A multiindex sum is to be written in the following way:

$\sum _{\genfrac{}{}{0ex}{}{\varrho \in {\mathbb{N}}_0^d}{\varrho \le \alpha }}$

Sources

• Refer to all the books and articles you take information from; generously refer to the work of others. The sources should be compiled at the end of each page (the term 'page' refers here to 'HTML-Web' page, and not printed page or monitor page).