Partial Differential Equations/Stylistic guidelines

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

Language[edit]

  • 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]

  • Only leave trivial things to the reader.
  • Put complicated and very technical results into the appendix.

Theorems[edit]

  • Always mention the weaknesses of theorems.

Structure[edit]

  • 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.

Figures[edit]

  • 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]

  • 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, ...):
    • letter for generic element of a set: x
    • letters for vector constants: \mathbf u, \mathbf v, \mathbf w
    • 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, \varrho
    • letters for Schwartz functions: \phi, \theta
    • letter for sets not assumed to be open or closed: B
    • letters for open sets: O, U
    • letter for closed sets: A
    • letter for domains: \Omega
    • letter for compact sets: K
    • letters for convex sets: C, Q
    • letter for generic set: X
    • 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
    • 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:
      • Poisson's equation: P
      • Heat equation: E
      • Helmholtz' equation: Z
    • letters for generic natural number and summation indices: n, k, j
    • letters for sequence indices: l, m
    • imaginary unit: i
    • Euler's constant: e
    • letter for linear functions: L
    • 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 and parameter: G_\kappa
    • sets defined by conditions: \{x \in \text{a set} | x \text{ satisfies a condition} \}
    • element in index set: \upsilon \in \Upsilon
  • In arguments of solutions of time-dependent partial differential equations, write the time variable first and then the space variable.

Sources[edit]

  • 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).