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

Chapter 12: Functions of Several Variables and Differentiation

<a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0073451342/295036/ch12.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"> (21.0K)</a>Few things in baseball are as exciting as a home run. In the summer of 2004, Barry Bonds chased the all-time home run record held by Hank Aaron. Every time Bonds hit the ball, fans watched in anxious anticipation as the ball reached its peak height and then slowly dropped back to the field. Would the ball clear the fence and stay fair for a home run? Through years of experience, players can usually tell exactly where the ball will land. However, since baseballs do not follow simple parabolic paths, most spectators must wait for the ball to land to see whether a given fly ball is a home run.

Think for a minute about the factors that determine how far the ball goes. In our studies of projectile motion, we identified three forces that affect the path: gravity, drag and the Magnus force. If we know the initial velocity (both speed and angle) and initial spin, we can write down a differential equation whose solution closely approximates the flight of the ball. This gives us distance as a function of speed, angle and spin. In this chapter, we introduce some of the basic techniques needed to analyze functions of two or more variables. While many of the ideas are familiar, the details change as we move from one to two or more variables.

You have probably realized that our situation is really far more complicated than outlined here. Air drag depends on environmental factors such as temperature and humidity. Other factors include the type of pitch thrown, the wind velocity and the type of bat used.With all of these factors, we would need a function of ten or more variables! Fortunately, the calculus of functions of ten variables is very similar to the calculus of functions of two or three variables. The theory presented in this chapter is easily extended to as many variables as are needed in a particular application.

After studying the basic calculus for functions of several variables, you should be able to find extrema of relatively simple functions. Perhaps more importantly, you should understand enough about such functions to be able to approximate extrema of more complicated functions. Of course, in real applications, you are rarely given a convenient formula. Even so, the understanding of multivariable calculus that you develop here will help you to make sense of a broad range of complex phenomena.

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