# Partial Differential Equations/Stylistic guidelines

Appearance

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

### Language

[edit | edit source]- 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

[edit | edit source]- 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

[edit | edit source]- Always mention the weaknesses of theorems.

### Structure

[edit | edit source]- 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

[edit | edit source]- Include as many links to other Wikimedia pages as possible
- Do not link to unofficial/commercial pages or unethical journals

### Figures

[edit | edit source]- 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

[edit | edit source]- 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

[edit | edit source]- 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).