# 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 $\mathbb {R} ^{d}$ 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:
          $\sum _{{\scriptstyle \varrho \in \mathbb {N} _{0}^{d}} \atop {\scriptstyle \varrho \leq \alpha }}$

### 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).