# Partial Differential Equations/Stylistic guidelines

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.

• 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: ${\displaystyle x}$
• letters for vectors of generic vector space (for a generic vector in ${\displaystyle \mathbb {R} ^{d}}$ please use ${\displaystyle x}$ and ${\displaystyle y}$, see below at the notation for the spatial variable): ${\displaystyle \mathbf {u} }$, ${\displaystyle \mathbf {v} }$, ${\displaystyle \mathbf {w} }$
• letters for vector constants: ${\displaystyle \mathbf {b} }$, ${\displaystyle \mathbf {c} }$
• letters for solutions of pde's: ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$
• letter for a smooth function ${\displaystyle B\to \mathbb {R} }$ in linear partial differential equations: ${\displaystyle a}$
• letters for constants which are elements of a field: ${\displaystyle b,c}$
• letter for element of ${\displaystyle [0,1]}$: ${\displaystyle \lambda }$
• letter for spatial dimension: ${\displaystyle d}$
• letters for bump functions: ${\displaystyle \varphi }$, ${\displaystyle \vartheta }$
• letters for Schwartz functions: ${\displaystyle \psi }$, ${\displaystyle \theta }$
• letter for sets not assumed to be open or closed: ${\displaystyle S}$
• letters for open sets: ${\displaystyle O}$, ${\displaystyle U}$
• letter for closed sets: ${\displaystyle A}$
• letter for domains: ${\displaystyle \Omega }$
• letter for compact sets: ${\displaystyle C}$
• letter for convex sets: ${\displaystyle Q}$
• letter for generic set: ${\displaystyle X}$
• letter for metric space: ${\displaystyle M}$
• letter for generic vector space: ${\displaystyle V}$
• letter for topology: ${\displaystyle \tau }$
• letter for generic topological space: ${\displaystyle {\mathcal {X}}}$
• letter for generic topological vector space: ${\displaystyle {\mathcal {V}}}$
• letter for generic function: ${\displaystyle f}$
• letter for function of inhomogenous problems: ${\displaystyle f}$ (since this is the convention in many sources)
• letter for diffeomorphism: ${\displaystyle \psi }$
• letter for outward normal vector: ${\displaystyle \nu }$
• letter for hessian matrix of ${\displaystyle f\in {\mathcal {C}}^{2}(O)}$: ${\displaystyle H_{f}}$
• letters for initial/boundary conditions: ${\displaystyle g}$, ${\displaystyle h}$
• letter for auxiliary function (and its variable): ${\displaystyle \mu (\xi )}$
• letter for curve (and its variable): ${\displaystyle \gamma (\rho )}$
• letters for vector fields: ${\displaystyle \mathbf {V} }$, ${\displaystyle \mathbf {W} }$
• letters for multiindices: ${\displaystyle \alpha }$, ${\displaystyle \beta }$, ${\displaystyle \varrho }$, ${\displaystyle \varsigma }$
• Priority: Generic multiindex in that order, summation index in reversed order
• letters for time and space: ${\displaystyle t}$, ${\displaystyle x}$ (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: ${\displaystyle s}$, ${\displaystyle y}$
• tertiary letter for space: ${\displaystyle z}$ (unfortunately, but there is no other suitable candidate)
• letter for radius: ${\displaystyle R}$
• notation for area and volume of ${\displaystyle d}$-dimensional sphere with radius ${\displaystyle R}$: ${\displaystyle A_{d}(R)}$, ${\displaystyle V_{d}(R)}$
• letter for generic fundamental solution: ${\displaystyle F}$
• notation for Green's kernels:
• Generic green's kernel: ${\displaystyle K}$
• Green's function: ${\displaystyle G}$
• Poisson's equation: ${\displaystyle P}$
• Heat equation: ${\displaystyle E}$
• Helmholtz' equation: ${\displaystyle Z}$
• letters for generic natural number and summation indices: ${\displaystyle n,k,j}$
• Priority: For summation ${\displaystyle j,k,n}$, for generic natural number ${\displaystyle n,k,j}$
• letters for sequence indices: ${\displaystyle l,m}$
• letters for natural numbers above which something holds: ${\displaystyle N,J,M}$
• notation for ${\displaystyle d}$-dimensional multiindex consisting only of ${\displaystyle l}$s: ${\displaystyle \varrho (d,l)}$
• imaginary unit: ${\displaystyle i}$
• Euler's constant: ${\displaystyle e}$
• letter for linear functions: ${\displaystyle T}$
• fundamental lagrange polynomial: ${\displaystyle \ell _{k,x_{1},\ldots ,x_{n}}}$
• Interpolating polynomial: ${\displaystyle L_{f,x_{1},\ldots ,x_{n}}}$
• letter for linear and continuous functions: ${\displaystyle {\mathcal {L}}}$
• letter for members of a dual space: ${\displaystyle {\mathcal {T}}}$ (for regular (tempered) distributions generated by ${\displaystyle f}$: ${\displaystyle {\mathcal {T}}_{f}}$)
• letter for the Gaussian function: ${\displaystyle \phi }$
• sets defined by conditions: ${\displaystyle \{x\in {\text{a set}}|x{\text{ satisfies a condition}}\}}$
• element in index set: ${\displaystyle \upsilon \in \Upsilon }$
• letter for set of continuous functions: ${\displaystyle {\mathcal {Q}}}$
• 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:
${\displaystyle \sum _{{\scriptstyle \varrho \in \mathbb {N} _{0}^{d}} \atop {\scriptstyle \varrho \leq \alpha }}}$

### 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).