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Unsuspected dangers of extrapolating from truncated analyses

Akhlesh Lakhtakia and Richard S. Andrulis Jr.

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?

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Unsuspected dangers of extrapolating from truncated analyses

Akhlesh Lakhtakia and Richard S. Andrulis Jr.

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?

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Publish Date: 01 June 1991


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