The Most Beautiful Equations in Applied Mathematics


From p. 171 of PCAM, typeset in all its splendour in the Lucida Bright font.

The BBC Earth website has just published a selection of short articles on beautiful mathematical equations and is asking readers to vote for their favourite.

I wondered if we had included these equations in The Princeton Companion to Applied Mathematics (PCAM), specifically in Part III: Equations, Laws, and Functions of Applied Mathematics. We had indeed included the ones most relevant to applied mathematics. Here are those equations, with links to the BBC articles.

  • The wave equation (which quotes PCAM author Ian Stewart). PCAM has a short article by Paul Martin of the same title (III.31), and the wave equation appears throughout the book.
  • Einstein’s field equation. PCAM has a 2-page article Einstein’s Field Equations (note the plural), by Malcolm MacCallum (article III.10).
  • The Euler-Lagrange equation. PCAM article III.12 by Paul Glendinning is about these equations, and more appears in other articles, especially The Calculus of Variations (IV.6), by Irene Fonseca and Giovanni Leoni.
  • The Dirac equation. A 3-page PCAM article by Mark Dennis (III.9) describes this equation and its quantum mechanics roots.
  • The logistic map. PCAM article The logistic equation (III.19), by Paul Glendinning treats this equation, in both differential and difference forms. It occurs in several places in the book.
  • Bayes’ theorem. This theorem appears in the PCAM article Bayesian Inference in Applied Mathematics (V.11), by Des Higham, and in other articles employing Bayesian methods.

A natural equation is: Are there other worthy equations that are the subject of articles in Part III of PCAM that have not been included in the BBC list? Yes! Here are some examples (assuming that only single equations are allowed, which rules out the Cauchy-Riemann equations, for example).

  • The Black-Scholes equation.
  • The diffusion (or heat) equation.
  • Laplace’s equation.
  • The Riccati equation.
  • Schrödinger’s equation.
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