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?

Access to the full text of this article is restricted. In order to view this article please log in.

Add a Comment