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

896 Chapter 27 Line integrals and multiple integrals
surface S
open surface S
dS
volume V
outwardly drawn normal
dS
Figure27.23
The divergence theoremrelates avolume integral
to asurface integral.
bounding curve C
Figure27.24
An open surfaceSwith bounding
curveC.
27.8.2 Stokes’theorem
Stokes’ theorem relates the integral over an open surface,S, to a line integral around a
closed curve,C, which bounds thatsurfacesuch as thatshown inFigure 27.24.
When a surface is open we adopt the following convention when drawing vectors
normal to the surface. When the direction of the normal vector has been specified, use
theright-handscrewruletoobtainasenseofturningaroundthenormalvector,asshown
inFigure27.24.Imaginenowmovingthecirclewhichsurroundsthenormalvectoralong
the surface until it just meets the curveC. Transfer its sense of turning to the curveC.
WhencalculatingthelineintegralaroundthecurveC,itshouldbetraversedinthesame
sense.
SpecificallyconsiderthetwoopensurfacesshowninFigure27.25.Inbothcaseswe
are considering cubes. In the first case the cube has no top face. In the second case the
cubehasnobottomface.Drawingvectordxdzjnotethatthesenseofturningrequiredby
theright-handscrewruleisthatshown.ThecurveC mustbetraversedinthedirections
shown.
Recall thatwhen a curve isclosed the symbol foralineintegral is ∮ C .
z
z
dS = dx dy k
C
dS=–dxdzj
dS=dxdzj
dS=–dxdzj
dS=dxdzj
y
y
C
x
x
dS= –dxdyk
Figure27.25
The right-handscrew rulegivesthe directionin whichC must be traversed.