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# American Mathematical Monthly - January 1997

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**Click on the months above to see summaries of articles in the MONTHLY. **

An archive for all the 1997 issues is now available

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Articles

**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 *P**c* 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.

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