Functions
what functions that we have to learn when we get older?
Dear Tara, As you get older, you need to learn the functions that will help you to do whatever it is that you want to accomplish. The basic functions are encountered so often that they are considered fundamental: linear (F(x) = mx+b, polynomial (F(x) = ax + bx squared, etc.), trigonometric (sine x, cosine x, etc.), and exponential functions. These functions are used to describe the behavior of many things that happen in the real world, from economics to biology and physics. For example, the behavior of a rubber band, a swing, the beams in a skyscraper being hit by a strong wind, water waves and a guitar string are all described reasonably well by trigonometric functions. Trigonometric functions are solutions to an equation called the wave equation, which describes many different situations. Many fields also use specialized functions because they describe the answer to a commonly encountered problem in that field. For example, the ripples that you see in the top of a Styrofoam coffee cup when you pull it gently across a slightly damp counter are described by a group of functions called Bessel functions. Try this, it's a great excuse for playing with your food. Drag a Styrofoam cup of coffee across a smooth, slightly damp counter. You'll see the surface of the cup break into a bunch of little ripples. It works best with some milk in the coffee; you can see the ripples better and the milk helps keep the ripples from breaking up. You're seeing higher order Bessel functions. Bessel functions are also solutions to the wave equation. When the physical object resembles a line (guitar strings, skyscraper beams) or can be thought of as having no boundaries (ocean waves, for example), the solutions to the wave equation are sines and cosines. When the object has a circular boundary (drum heads, coffee cups, flow in pipes, light in a fiber optic cable and many more), the solutions are a group of functions called Bessel functions. You can see J0, the first Bessel function, just by tapping a drum in the center, The whole head goes up and down together. Now tap it about a third of the way from the edge; one half will go up while the other half goes down. That's J11. If you tap carefully, you can get more: the center of the drum head goes down while the edge goes up (J2) or pie shaped sections vibrate up and down (J21 or higher). High order Bessel functions are combinations of circular and pie-shaped vibrations. Those are the ripples in the coffee cup, where the surface breaks up into a lot of little bumps, each vibrating separately up and down. There are many other functions that are useful for describing particular situations. Population biology uses a whole collection of functions to describe the growth and interaction of populations. These functions are also useful in related fields such as ecology and sociology but seldom occur in physics. Economic uses some of the same functions as physics, but also some specialized ones. So Tara, after you know the basic functions, the additional ones that you will need depend on the field that you want to study. Regards, Celt M. Schira, P.E.