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:
 * $$\sum_{{\scriptstyle \varrho \in \mathbb N_0^d} \atop {\scriptstyle \varrho \le \alpha}}$$