# Partial Differential Equations/Stylistic guidelines

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

## Contents

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