Surface Integral

For a Scalar Function $f$ over a surface parameterized by $u$ and $v$, the surface integral is given by

$$\Phi = \int_S f \,da = \int_S f(u,v)\, \vert{\bf T}_u\times {\bf T}_v\vert\,du \,dv,$$ (1)

where ${\bf T}_u$ and $\hat{\bf T}_v$ are tangent vectors and ${\bf a}\times{\bf b}$ is the Cross Product.

For a Vector Function over a surface, the surface integral is given by

$$\Phi = \int_S {\bf F}\cdot d{\bf a} = \int_S ({\bf F}\cdot \hat {\bf n})\,da$$ (2)

$$= \int_S f_x\,dy\,dz + f_y\,dz\,dx + f_z\,dx\,dy,$$ (3)

where ${\bf a}\cdot{\bf b}$ is a Dot Product and $\hat{\bf n}$ is a unit Normal Vector. If $z = f(x,y)$, then $d{\bf a}$ is given explicitly by

$$d{\bf a} = \pm \left({-{\partial z\over\partial x} \hat{\bf ... ...l z\over\partial y} \hat{\bf y} + \hat{\bf z}}\right)\,dx\,dy.$$ (4)

If the surface is Surface Parameterized using $u$ and $v$, then

$$\Phi = \int_S {\bf F}\cdot ({\bf T}_u\times {\bf T}_v)\,du \,dv.$$ (5)

See also Surface Parameterization