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American Mathematical Monthly - January 1997

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An archive for all the 1997 issues is now available


Reading the Master: Newton and the Birth of Celestial Mechanics Bruce Pourciau


Based on a mere two-sentence sketch in the Principia, historians of science credit Newton with the first proof that every orbital motion subject to an inverse-square force must lie on a conic. Against a backdrop of historical anecdote, this article reveals how Newton's sketch unfolds into a convincing proof and then compares this original proof with a contemporary demonstration using vector calculus. Newton's original argument calls on the converse -- that conic motions about the focus must have inverse-square accelerations -- and the article goes on to study the Principia's very geometric proof of this reversal


Applications of Linear Algebra in Calculus
Jack W. Rogers, Jr.


Two applications of linear algebra in calculus are used to help motivate the concepts of the matrix for a linear transformation between finite dimensional vector spaces and of a change-of-basis matrix. The first application uses the concept of the matrix for the differentiation transformation D: C^1 (R) -> C (R), restricted to an appropriate finite-dimensional subspace, to show how integration by parts can often be replaced by matrix inversion. The second application uses the concept of a change-of-basis matrix to clarify the integration of polynomials in the sine and cosine. For example, the bases B = (1, cos t, ..., cos^n t) and C = (1, cos t, ..., cos nt) span the same linear space. The change-of-basis matrix Pc B converts the problem of integrating powers of the cosine into the much simpler one of investigating elements of C.

Applications of Linear Algebra in Calculus
John A. Baker


For a continous, real-valued function f on S^n-1 (the set of all unit vectors in IR^n), the "surface: integral of f over S^n-1 can be defined by
From this unconventional definition, an n-dimensional analogue of the "polar coordinates change of variable theorem" follows with relative ease. Mixing in some well-cured properties of Euler's gamma function, we compute the integral of a monomial over S^n-1 and thereby deduce the divergence theorem for polynomial fields over the unit ball in IR^n. The Weierstrass approximation theorem is then used to complete the proof of the divergence theorem for balls in IR^n.

Math Lingo vs. Plain English: Double Entendre
Reuben Hersh


This is a collection of examples where the same words have different meanings in math talk and plain English talk. It is intended as a cautionary tale for math teachers.