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
highspeed computers were available, engineers built smallscale
or fullscale 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
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CHAPTER 14
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Vector Calculus
142
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 threedimensional 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 lowslung (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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 Spring '10
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 Derivative, Volkswagen, Vector field

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