Surface integral

Illustration of a surface formed by a collection of surface elements

A surface integral is a double integral applied over a surface. It is, in some ways, the higher-dimensional equivalent of a line integral. A surface integral can be applied over a scalar or vector field, the latter of which is sometimes called a flux integral.

Types of surface integrals

Over a scalar field

In its most basic form, a surface integral of $f(x,y,z)$ over the surface $g(x,y)$ takes the form

$\iint _{S}f(x,y,z)dS=\iint _{D}f{\big (}x,y,g(x,y){\big )}{\sqrt {1+\left({\frac {\partial g}{\partial x}}\right)^{2}+\left({\frac {\partial g}{\partial y}}\right)^{2}}}dA$

where $S$ is the surface and $D$ is the projection of $S$ onto the $xy$ plane. Note that if $f(x,y,z)=1$ , this integral becomes the surface area of $g(x,y)$ .

The more general form is found by parameterizing the surface $g$ to a vector function $r(u,v)$ , in which case it would become:

$\iint _{S}f(x,y,z)dS=\iint _{S}f({\vec {r}}){\big \|}{\vec {r}}_{u}\times {\vec {r}}_{v}{\big \|}dA$

Over a vector field

Over a vector field, a surface or flux integral is equal to

$\iint _{S}{\vec {F}}\cdot d{\vec {r}}$	$=\iint _{S}{\vec {F}}\cdot {\hat {n}}\,dS=\iint _{S}{\vec {F}}({\vec {r}})\cdot \pm {\frac {{\vec {r}}_{u}\times {\vec {r}}_{v}}{{\big \|}{\vec {r}}_{u}\times {\vec {r}}_{v}{\big \|}}}dS$
	$=\iint _{S}{\vec {F}}({\vec {r}})\cdot \pm ({\vec {r}}_{u}\times {\vec {r}}_{v})dA$

where ${\vec {F}}$ is a vector-valued function, ${\hat {n}}$ is the unit normal vector to a surface, and ${\vec {r}}$ is the vector-valued parametric representation of the surface being integrated over. In the special case that the surface can be represented as $z=g(x,y)$, it can be shown that this formula reduces to

$\iint _{S}{\vec {F}}\cdot d{\vec {S}}=\iint _{D}\left(-F_{x}{\frac {\partial g}{\partial x}}-F_{y}{\frac {\partial g}{\partial y}}+F_{z}\right)dA,$

where D is the projection of S onto the xy-plane.

Whether the value is positive or negative is arbitrary depending on which was the unit normal vectors should be pointing. By convention, for any closed surface, the normal vectors point outwards.