Page 246 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 10] LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS 237

Fig. 10-5

ðð ð ðð
@A 1 @A 2 @A 3
ðA 1 cos þ A 2 cos þ A 3 cos
Þ dS
@x þ @y þ @z dV ¼ ð24Þ
S
The rectangular Cartesian component form of (23)is
ððð ðð
@A 1 @A 2 @A 3
@x þ @y þ @z dV ¼ ðA 1 dy dz þ A 2 dz dx þ A 3 dx dyÞ ð25Þ
V S

EXAMPLE. If B is the magnetic field vector, then one of Maxwell’s equations of electromagnetic theory is
r B ¼ 0. When this equation is substituted into the left member of (23), the right member tells us that the
magnetic flux through a closed surface containing a magnetic field is zero. A simple interpretation of this fact
results by thinking of a magnet enclosed in a ball. All magnetic lines of force that flow out of the ball must return
(so that the total flux is zero). Thus, the lines of force flow from one pole to the other, and there is no dispersion.

STOKES’ THEOREM

Stokes’ theorem establishes the equality of the double integral of a vector field over a portion of a
surface and the line integral of the field over a simple closed curve bounding the surface portion. (See
Fig. 10-6.)
Suppose a closed curve, C, bounds a smooth surface portion, S. If the component functions of
x ¼ rðv 1 ; v 2 Þ have continuous mixed partial derivatives, then for a vector field A with continuous partial
derivatives on S
þ ðð
n r A dS
A dr ¼ ð26Þ
C
S
where n ¼ cos i þ cos j þ cos
k with ; , and
representing the angles made by the outward normal
n and i; j, and k, respectively.
Then the component form of (26)is
þ ðð
@A 3 @A 2 @A 1 @A 3 @A 2 @A 1
cos
dS
ðA 1 dx þ A 2 dy þ A 3 dzÞ¼ cos þ cos þ
C @y @z @z @x @x @y
S
ð27Þ
þ
If r A ¼ 0, Stokes’ theorem tells us that A dr ¼ 0. This is Theorem 3 on Page 237.
C