For a scalar function over a surface parameterized by and , the surface integral is given by
(1)
| |||
(2)
|
where and are tangent vectors and is the cross product.
For a vector function over a surface, the surface integral is given by
(3)
| |||
(4)
| |||
(5)
|
where is a dot product and is a unit normal vector. If , then is given explicitly by
(6)
|
If the surface is surface parameterized using and , then
(7)
|