The Calculus Diaries - Jennifer Ouellette [41]
De Brosses admired her intellectual prowess greatly, and was horrified upon learning that she wished to become a nun. She did become a nun, but not before spending ten years writing a seminal mathematics textbook, Analytical Institutions, published in 1748—the first surviving mathematical treatise written by a woman. She was also the first woman to be appointed a mathematics professor at a university (the University of Bologna), although there is no record she ever formally accepted the position. She died a pauper in 1799, having given away everything she owned.
But her work lives on. One of the curves featured in Analytical Institutions is the Witch of Agnesi. Agnesi dubbed it la versiera, a nautical term meaning “a rope that turns a sail”—an allusion to the motion by which the curve is drawn.
At some point, a harried English translator misinterpreted the word as l’avversiera, “she-devil” or “witch.”
What does this have to do with the pendulum motion of the Devil Dive? Among other things, this curve describes a driven oscillator near resonance—a swinging pendulum that is being poked or prodded to keep it in motion, for example, like someone pushing a child on a swing. When the rate of prodding matches the rate of the pendulum’s swing, it is said to be in resonance. If the rate of prodding is very, very close to the rate of the swing, the amplitude (height) of the swing, plotted as a function of frequency, forms the Witch of Agnesi. We’ve already seen that the physical motion of a pendulum forms an arc, while plotting its position as a function of time gives us a periodic sine wave. So had someone (or something) been pushing us during the pendulum phase of the ride at almost the exact same rate as our swing, the Witch of Agnesi would describe our amplitude as a function of forced frequency (rate of prodding).
V IS FOR VECTOR
Making our way into Fantasyland, we find the King Arthur Carousel and the Dumbo-inspired flying-elephant ride, both excellent examples of rotation around a fixed axis. But it is the Mad Tea Party—usually called the spinning teacup ride—that provides us with an unusual illustration of vectors: motion in specific directions. A vector is technically defined as any quantity having both direction and magnitude. In physics, vectors typically describe force, velocity, acceleration, or similar three-dimensional properties. How the different vectors combine determines their net strength; one must take into account not just how strong a given force might be, but also in what direction it is pushing.
It is easiest to illustrate the concept in one dimension. Picture your standard number line. An object moving in a straight line has a direction, depicted by a small arrow above the number. If it starts at 0 and ends at 5, this is called vector (5); it’s the same as any other number along that line, except we have specified a direction. Because it’s moving left to right, it is a positive number. A vector pointing from right to left would be a negative number. Vectors can be added together or subtracted, just like regular numbers. Combine vector (5) with vector (−5), and the two cancel each other out completely; combine it with vector (−3), and you end up with vector (2); combine it with vector (4), and you end up with vector (9). And so on.
Frankly, vectors aren’t very interesting in the one-dimensional realm of the number line: There is no real difference between them and ordinary numbers. In two dimensions, vectors are pairs of numbers (Cartesian coordinates) that describe the direction of movement in a plane. In three dimensions, they describe directional motion through space using three coordinates.
Here is how vectors apply to the Mad Tea Party. Any rotating body’s motion has a vector that is constantly changing, because the direction shifts at each point