Page 245 - Schaum's Outline of Theory and Problems of Advanced Calculus

236 LINE INTEGRALS, SURFACE INTEGRALS, AND INTEGRAL THEOREMS [CHAP. 10

where ð p ; p ; p Þ is some point of S p . If the limit of this sum as n !1 in such a way that each
S p ! 0 exists, the resulting limit is called the surface integral of ðx; y; zÞ over S and is designated by
ðð
ðx; y; zÞ dS ð18Þ
S
Since S p ¼j sec
p j A p approximately, where
p is the angle between the normal line to S and the
positive z-axis, the limit of the sum (17)can be written
ðð
ðx; y; zÞj sec
j dA ð19Þ
r
The quantity j sec
j is given by
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2
1 @z @z
@x @y
j sec
j¼ ¼ 1 þ þ ð20Þ
jn p kj
Then assuming that z ¼ f ðx; yÞ has continuous (or sectionally continuous) derivatives in r,(19) can be
written in rectangular form as
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðð 2 2
@z @z
dx dy
@x @y
ðx; y; zÞ 1 þ þ ð21Þ
r
In case the equation for S is given as Fðx; y; zÞ¼ 0, (21) can also be written
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 2 2
ðð
ðF x Þ þðF y Þ þðF z Þ
dx dy
ðx; y; zÞ ð22Þ
jF z j
r
The results (21)or (22) can be used to evaluate (18).
In the above we have assumed that S is such that any line parallel to the z-axis intersects S in only
one point. In case S is not of this type, we can usually subdivide S into surfaces S 1 ; S 2 ; ... ; which are of
this type. Then the surface integral over S is defined as the sum of the surface integrals over S 1 ; S 2 ; ... .
The results stated hold when S is projected on to a region r on the xy plane. In some cases it is
better to project S on to the yz or xz planes. For such cases (18) can be evaluated by appropriately
modifying (21) and (22).

THE DIVERGENCE THEOREM
The divergence theorem establishes equality between triple integral (volume integral) of a function
over a region of three-dimensional space and the double integral of the function over the surface that
bounds that region. This relation is very important in the expression of physical theory. (See Fig.
10-5.)

Divergence (or Gauss) Theorem
Let A be a vector field that is continuously differentiable on a closed-space region, V, bound by a
smooth surface, S. Then
ðð ð ðð
A n dS
r A dV ¼ ð23Þ
V S
where n is an outwardly drawn normal.
If n is expressed through direction cosines, i.e., n ¼ i cos þ j cos þ k cos
, then (23) may be
written