Partial Differential Equations/Stylistic guidelines

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


  • 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.


  • 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.


  • Always mention the weaknesses of theorems.


  • 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]

  • Include as many links to other Wikimedia pages as possible
  • Do not link to unofficial/commercial pages or unethical journals


  • 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.


  • 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:


  • 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).