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: letters for vectors of generic vector space (for a generic vector in please use and , see below at the notation for the spatial variable): , , letters for vector constants: , letters for solutions of pde's: , , letter for a smooth function in linear partial differential equations: letters for constants which are elements of a field: letter for element of : letter for spatial dimension: letters for bump functions: , letters for Schwartz functions: , letter for sets not assumed to be open or closed: letters for open sets: , letter for closed sets: letter for domains: letter for compact sets: letter for convex sets: letter for generic set: letter for metric space: letter for generic vector space: letter for topology: letter for generic topological space: letter for generic topological vector space: letter for generic function: letter for function of inhomogenous problems: (since this is the convention in many sources) letter for diffeomorphism: letter for outward normal vector: letter for hessian matrix of : letters for initial/boundary conditions: , letter for auxiliary function (and its variable): letter for curve (and its variable): letters for vector fields: , letters for multiindices: , , , Priority: Generic multiindex in that order, summation index in reversed order letters for time and space: , (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: , tertiary letter for space: (unfortunately, but there is no other suitable candidate) letter for radius: notation for area and volume of -dimensional sphere with radius : , letter for generic fundamental solution: notation for Green's kernels: Generic green's kernel: Green's function: Poisson's equation: Heat equation: Helmholtz' equation: letters for generic natural number and summation indices: Priority: For summation , for generic natural number letters for sequence indices: letters for natural numbers above which something holds: notation for -dimensional multiindex consisting only of s: imaginary unit: Euler's constant: letter for linear functions: fundamental lagrange polynomial: Interpolating polynomial: letter for linear and continuous functions: letter for members of a dual space: (for regular (tempered) distributions generated by : ) letter for the Gaussian function: sets defined by conditions: element in index set: letter for set of continuous functions: 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: 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).