Early Transcendental Functions (Smith-Minton), 3rd Edition

Chapter 9: Parametric Equations and Polar Coordinates

<a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0073451342/295036/ch09.jpg','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (22.0K)</a>

You are all familiar with sonic booms, those loud crashes of noise caused by aircraft flying faster than the speed of sound. You may have even heard a sonic boom, but you have probably never seen a sonic boom. The remarkable photograph here shows water vapor outlining the surface of a shock wave created by an F-18 jet flying supersonically. (Note that there is also a small cone of water vapor trailing the back of the cockpit of the jet.) <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0073451342/295036/ch09_sonic_boom.jpg','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (6.0K)</a> You may be surprised at the apparently conical shape assumed by the shock waves. A mathematical analysis of the shock waves verifies that the shape is indeed conical. (You will have an opportunity to explore this in the exercises in section 9.1.) To visualize how sound waves propagate, imagine an exploding firecracker. If you think of this in two dimensions, you'll recognize that the sound waves propagate in a series of ever-expanding concentric circles that reach everyone standing a given distance away from the firecracker at the same time. In this chapter, we extend the concepts of calculus to curves described by parametric equations and polar coordinates. For instance, in order to study the motion of an object such as an airplane in two dimensions, we would need to describe the object's position (x, y) as a function of the parameter t (time). That is, we write the position in the form (x, y) = (x(t), y(t)), where x(t) and y(t) are functions to which our existing techniques of calculus can be applied. The equations x = x(t) and y = y(t) are called parametric equations. Additionally, we'll explore how to use polar coordinates to represent curves, not as a set of points (x, y), but rather, by specifying the points by the distance from the origin to the point and an angle corresponding to the direction from the origin to the point. Polar coordinates are especially convenient for describing circles, such as those that occur in propagating sound waves. These alternative descriptions of curves bring us a great deal of needed flexibility in attacking many problems. Often, even very complicated looking curves have a simple description in terms of parametric equations or polar coordinates. We explore a variety of interesting curves in this chapter and see how to extend the methods of calculus to such curves.