Page 243 - Schaum's Outline of Theory and Problems of Advanced Calculus

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

Fig. 10-3

(This is the usual geometric interpretation of the cross product abstracted to the differential level.)
This strongly suggests the following definition:

Definition. The differential element of surface area is

@r
@r

dS ¼ dv 1 dv 2 ð12Þ

@v 1 @v 2
For a function ðv 1 ; v 2 Þ that is everywhere integrable on S
ðð ðð
@r
@r

dS ¼ ðv 1 ; v 2 Þ dv 1 dv 2 ð13Þ

@v 1 @v 2
S S
is the surface integral of the function :
In general, the surface integral must be referred to three-space coordinates to be evaluated. If the
surface has the Cartesian representation z ¼ f ðx; yÞ and the identifications
v 1 ¼ x; v 2 ¼ y; z ¼ f ðv 1 ; v 2 Þ
are made then
@r @z @r @z
k; k
¼ i þ ¼ j þ
@v 1 @x @v 2 @y
and
@r @r @z @z
i
¼ k j
@v 2 @v 2 @y @x
Therefore,
" # 1=2
2 2
@r @z @z
@r

¼ 1 þ þ
@x @y
@v 1 @v 2
Thus, the surface integral of has the special representation
" # 1=2
ðð 2 2
@z @z
dx dy
S ¼ ðx; y; zÞ 1 þ þ ð14Þ
@x @y
S
If the surface is given in the implicit form Fðx; y; zÞ¼ 0, then the gradient may be employed to
obtain another representation. To establish it, recall that at any surface point P the gradient, rF is
perpendicular (normal) to the tangent plane (and hence to S).
Therefore, the following equality of the unit vectors holds (up to sign):