# 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.

• 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 $\mathbb {R} ^{d}$  please use $x$  and $y$ , see below at the notation for the spatial variable): $\mathbf {u}$ , $\mathbf {v}$ , $\mathbf {w}$
• letters for vector constants: $\mathbf {b}$ , $\mathbf {c}$
• letters for solutions of pde's: $u$ , $v$ , $w$
• letter for a smooth function $B\to \mathbb {R}$  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: $\varphi$ , $\vartheta$
• 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: $\Omega$
• 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: ${\mathcal {X}}$
• letter for generic topological vector space: ${\mathcal {V}}$
• 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 $f\in {\mathcal {C}}^{2}(O)$ : $H_{f}$
• 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: $\mathbf {V}$ , $\mathbf {W}$
• 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: $\ell _{k,x_{1},\ldots ,x_{n}}$
• Interpolating polynomial: $L_{f,x_{1},\ldots ,x_{n}}$
• letter for linear and continuous functions: ${\mathcal {L}}$
• letter for members of a dual space: ${\mathcal {T}}$  (for regular (tempered) distributions generated by $f$ : ${\mathcal {T}}_{f}$ )
• letter for the Gaussian function: $\phi$
• sets defined by conditions: $\{x\in {\text{a set}}|x{\text{ satisfies a condition}}\}$
• element in index set: $\upsilon \in \Upsilon$
• letter for set of continuous functions: ${\mathcal {Q}}$
• 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:
$\sum _{{\varrho \in \mathbb {N} _{0}^{d}} \atop {\varrho \leq \alpha }}$

### 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).