Page 268 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 10] LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS 259

where V is the volume enclosed by the surface S and the limit is obtained by shrinking V to
the point P.
ððð ðð
By the divergence theorem, div A dV ¼ A n dS
V S
By the mean value theorem for integrals, the left side can be written
ðð ð
div A dV ¼ div A V
V
where div A is some value intermediate between the maximum and minimum of div A throughout V.
Then
ðð
A n dS
S
V
div A ¼
Taking the limit as V ! 0suchthat P is always interior to V, div A approaches the value div A at
point P;hence
ðð
A n dS
div A ¼ lim S
V!0 V
This result can be taken as a starting point for defining the divergence of A, and from it all the
properties may be derived including proof of the divergence theorem. We can also use this to extend the
concept of divergence to coordinate systems other than rectangular (see Page 159).
0 1
ððð
A n ds A = V represents the flux or net outflow per unit volume of the vector A from
Physically, @
S
the surface S.If div A is positive in the neighborhood of a point P,itmeans that the outflow from P is
positive and we call P a source.Similarly, if div A is negative in the neighborhood of P,the outflow is really
an inflow and P is called a sink.Ifina region there are no sources or sinks, then div A ¼ 0 and we call A a
solenoidal vector field.

Supplementary Problems

LINE INTEGRALS
ð
ð4;2Þ
2
10.35. Evaluate ðx þ yÞ dx þðy xÞ dy along (a)the parabola y ¼ x,(b)a straight line, (c) straight lines
ð1;1Þ
2
2
from ð1; 1Þ to ð1; 2Þ and then to ð4; 2Þ, (d)the curve x ¼ 2t þ t þ 1; y ¼ t þ 1.
Ans: ðaÞ 34=3; ðbÞ 11; ðcÞ 14; ðdÞ 32=3
þ
10.36. Evaluate ð2x y þ 4Þ dx þð5y þ 3x 6Þ dy around a triangle in the xy plane with vertices at ð0; 0Þ; ð3; 0Þ,
ð3; 2Þ traversed in a counterclockwise direction. Ans. 12
10.37. Evaluate the line integral in the preceding problem around a circle of radius 4 with center at ð0; 0Þ.
Ans: 64
ð
2
2
2
10.38. (a)If F ¼ðx y Þi þ 2xyj,evaluate F dr along the curve C in the xy plane given by y ¼ x x from the
C
point ð1; 0Þ to ð2; 2Þ. (b)Interpret physically the result obtained.
Ans. (a) 124/15