# The College Mathematics Journal

### March 2013 Contents

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The March issue of The College Mathematics Journal highlights "Women and Mathematics in the Time of Euler," Betty Mayfield's evocation of the mathematical life of women in the 18th century. Adam Parker investigates "Who Solved the Bernoulli Differential Equation and How Did They Do It?" which takes us back to the late 17th century invoking Leibniz and the Bernoulli brothers. Other articles present fresh perspectives on derivative sign patterns, the Tower of Hanoi problem, the Pythagorean theorem, and much more.

Women and Mathematics in the Time of Euler

Betty Mayfield

We explore mathematics written both by and for women in 18th-century Europe, and some of the interesting personalities involved: Maria Agnesi, Emilie du Châtelet, Laura Bassi, Princess Charlotte Ludovica Luisa, John Colson, Francesco Algarotti, and Leonhard Euler himself.

Who Solved the Bernoulli Differential Equation And How Did They Do It?

The Bernoulli brothers, Jacob and Johann, and Leibniz: any of these might have been first to solve what is called the Bernoulli differential equation. We explore their ideas here, and the chronology of their work, finding out, among other things, that variation of parameters was used in 1697, 78 years before 1775, when Lagrange introduced it in general.

An n-dimensional Pythagorean Theorem

William J. Cook

An n-dimensional generalization of the standard cross product, leads to an n-dimensional generalization of the Pythagorean theorem.

Derivative Sign Patterns in Two Dimensions

Ken Schilling

Given a function defined on a subset of the plane, whose partial derivatives never change sign, the signs of the partial derivatives form a two-dimensional pattern. We explore what patterns are possible for various planar domains.

Teaching Tip: When an Invertible Matrix and Its Inverse are Both Stochastic

J. Ding and N. H. Rhee

A stochastic matrix is a square matrix with nonnegative entries and row sums one. The simplest example is a permutation matrix, whose rows permute the rows of an identity matrix. A permutation matrix and its inverse are both stochastic. We prove the converse; that is, if a matrix and its inverse are both stochastic, then it is a permutation matrix.

Multi-Peg Tower of Hanoi

Paul Isihara and Doeke Buursma

A simple algorithm for multi-peg Tower of Hanoi is proven to generate optimal Frame-Stewart partitions.

An Ellipse Morphs to a Cosine Graph!

L. R. King

We produce a continuum of curves all of the same length beginning with an ellipse and ending with a cosine graph. The curves in the continuum are made by cutting and unrolling circular cones whose section is the ellipse; the initial cone is degenerate (it is the plane of the ellipse); the final cone is a circular cylinder. The curves of the continuum show the ellipse from the perspective of the intrinsic geometry of the various cones.

Mathematical Minute: Rotating a Function Graph

Daniel Bravo and Joseph Fera

Using calculus only, we find the angles you can rotate the graph of a differentiable function about the origin and still obtain a function graph. We then apply the solution to polynomials of odd and even degree.

Ryan Hoban

Many classical problems in elementary calculus use Euclidean geometry. This article takes such a problem and solves it in hyperbolic and in spherical geometry instead. The solution requires only the ability to compute distances and intersections of points in these geometries. The dramatically different results we obtain illustrate the effect curvature has on basic geometric objects.

Proof Without Words: A Variation on Thébault's First Problem

Purna Patel and Raymond Viglione

A visual proof of a new variation of Thébault's first problem.

CLASSROOM CAPSULES

Slouching in the Rain

Herb Bailey

A number of papers find the velocity that minimizes the wetness of a traveler caught in the rain. In this capsule we determine, in addition, the amount of forward bend (slouching) that enables the traveler to stay as dry as possible.

A New Ratio Test for Positive Monotone Series

Hongwei Chen

Combining D'Alembert's ratio test and Cauchy's condensation test, we present a new ratio test for any positive monotone series.

SOFTWARE REVIEW

SAGE: Open Source Mathematics Software System

reviewed by J. K. Denny

PROBLEMS AND SOLUTIONS

MEDIA HIGHLIGHTS