Surface Integral

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)

REFERENCES:

Leathem, J. G. Volume and Surface Integrals Used in Physics. Cambridge, England: University Press, 1905.