"Oriented" Surface Integral:
∫
∫
F ·
dS
= ∫ ∫ F ·
n dS =
∫ ∫ F ·
(ru
x rv)
/ | ru
x rv
| dS
S S S
and recall dS =
| ru
x rv
|dA
where | ru
x rv
| transforms the element of area, dS,
on the surface, S, to
the element of area, dA, on the uv-surface
in the domain, D.
So the surface integral
becomes note: dS = | ru x rv |dA
∫
∫
F · dS = ∫
∫
F · n dS = ∫
∫
F · (ru x rv)
/ | ru
x rv
| dS =
∫ ∫ F · (ru x rv)
dA
S S S
D
Note that the dot product, F ·
(ru
x rv)
, is a scalar function and therefore
it is
a scalar field. So the same two approaches, the direct approach and the
transformation approach, still apply to evaluate the scalar
form of surface
integrals starting with
this vector form of surface integral.
|