082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

890 Chapter 27 Line integrals and multiple integrals
Engineeringapplication27.4
Theelectricchargeenclosedinaregion
Suppose electric charge is distributed throughout a volume,V, and at any point has
charge density ρ(x,y,z). Charge density isascalar field.
Supposeweselectaverysmallportionhavingvolume δV andlocatedatP(x,y,z)
as shown inFigure 27.18. The charge inthisportion, δQ, isgiven by
δQ=ρδV
Ifwewishtocalculatethetotalchargeenclosedwithinthevolume,wemustsumall
such contributions from the entire volume. This is found by integrating throughout
the volume. We write thisas
∫
total charge,Q = ρdV
This isanother example of a volume integral.
volume V
V
volume dV
charge density
r(x, y,z)
P (x, y,z)
Figure27.18
The total charge enclosed is foundby
summing, orintegrating, throughout
the whole volume.
Engineeringapplication27.5
Fluidflowacrossasurface
Figure27.19(a)representsthemotionofabodyoffluidthroughoutaregion.Atany
pointP(x,y,z) fluid will be moving with a certain speed in a certain direction. So,
each small fluid element has a particular velocityv, which varies with position, that
isv = v(x,y,z). This isan example of a vector field, as discussedinSection 7.4.
surface S
v = v(x, y,z)
v = v(x, y,z)
P(x, y,z)
dS
small portion of surface dS
(a)
(b)
Figure27.19
Thevector fieldvrepresentsfluid velocity andSisan imaginary surface through whichthe
fluidflows.