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# ch14_01 - P1 OSO/OVY GTBL001-14-nal P2 OSO/OVY QC OSO/OVY...

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CHAPTER 14 Vector Calculus The Volkswagen Beetle was one of the most beloved and recogniz- able cars of the 1950s, 1960s and 1970s. So, Volkswagen’s decision to release a redesigned Beetle in 1998 created quite a stir in the automotive world. The new Beetle resembles the classic Beetle, but has been modernized to improve gas mileage, safety, handling and overall performance. The calculus that we introduce in this chapter will provide you with some of the basic tools necessary for designing and analyzing automobiles, aircraft and other types of complex machinery. Think about how you might redesign an automobile to improve its aerodynamic performance. Engineers have identiﬁed many im- portant principles of aerodynamics, but the design of a complicated structure like a car still has an element of trial and error. Before high-speed computers were available, engineers built small-scale or full-scale models of new designs and tested them in a wind tunnel. Unfortunately, such models don’t always provide ade- The old Beetle The new Beetle quate information and can be prohibi- tively expensive to build, particularly if you have 20 or 30 new ideas you’d like to try. With modern computers, wind tun- nel tests can be accurately simulated by sophisticated programs. Mathematical models give engineers the ability to thor- oughly test anything from minor modiﬁ- cations to radical changes. The calculus that goes into a com- puter simulation of a wind tunnel is beyond what you’ve seen so far. Such sim- ulations must keep track of the air velocity at each point on and around a car. A func- tion assigning a vector (e.g., a velocity vector) to each point in space is called a vector ﬁeld, which we introduce in sec- tion 14.1. To determine where vortices and turbulence occur in a ﬂuid ﬂow, you must compute line integrals, which are 1115

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1116 CHAPTER 14 . . Vector Calculus 14-2 discussed in sections 14.2 and 14.3. The curl and divergence, introduced in section 14.5, allow you to analyze the rotational and linear properties of a ﬂuid ﬂow. Other properties of three-dimensional objects, such as mass and moments of inertia for a thin shell (such as a dome of a building), require the evaluation of surface integrals, which we develop in section 14.6. The relationships among line integrals, surface integrals, double integrals and triple integrals are explored in the remaining sections of the chapter. In the case of the redesigned Volkswagen Beetle, computer simulations resulted in nu- merous improvements over the original. One measure of a vehicle’s aerodynamic efﬁciency is its drag coefﬁcient. Without getting into the technicalities, the lower its drag coefﬁcient is, the less the velocity of the car is reduced by air resistance. The original Beetle has a drag coefﬁcient of 0.46 (as reported by Robertson and Crowe in Engineering Fluid Me- chanics ). By comparison, a low-slung (and quite aerodynamic) 1985 Chevrolet Corvette has a drag coefﬁcient of 0.34. Volkswagen’s speciﬁcation sheet for the new Beetle lists a drag coefﬁcient of 0.38, representing a considerable reduction in air drag from the original
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ch14_01 - P1 OSO/OVY GTBL001-14-nal P2 OSO/OVY QC OSO/OVY...

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