Unsuspected dangers of extrapolating from truncated analyses

Differential calculus is central to the physical sciences. Analyses of phenomena described by Maxwell's equations, Navier's equation, the Navier-Stokes equations, or the Korteweg-deVries equation, etc., all need differential calculus. Limits are essential to differential calculus. Differentiation of a function f(x) about a point x0 as a casual perusal of calculus textbooks as well as Newton's notebooks will suffice to show,3 has a geometric interpretation in which the spread (xb - xa) of the range xa ≤ x0 ≤ xb is made increasingly small until it almost vanishes. Assuming that f(x) is continuous, this limiting process yields its derivative at x0. Or, does it?

Become a member of OSA or log in to view the full text of this article.

OSA Members get the full text of Optics & Photonics News, plus a variety of other member benefits.

Add a Comment

comments powered by Disqus