  in mathematics, a surface integral is a generalization of multiple integrals to integration over surfaces. it can be thought of as the double integral analog of the line integral. given a surface, one may integrate over its scalar fields (that is, functions which return scalars as values), and vector fields (that is, functions which return vectors as values). surface integrals have applications in physics, particularly with the classical theory of electromagnetism. right|thumb|the definition of surface integral relies on splitting the surface into small surface elements. right|thumb|an illustration of a single surface element. these elements are made infinitesimally small, by the limiting process, so as to approximate the surface. == surface integrals of scalar fields == to find an explicit formula for the surface integral, we need to parameterize the surface of interest, s, by considering a system of curvilinear coordinates on s, like the latitude and longitude on a sphere. let such a parameterization be x(s, t), where (s, t) varies in some region t in the plane. then, the surface integral is given by : \iint_{s} f \,\mathrm ds = \iint_{t} f(\mathbf{x}(s, t)) \left\|{\partial \mathbf{x} \over \partial s}\times {\partial \mathbf{x} \over \partial t}\right\| \mathrm ds\, \mathrm dt where the expression between bars on the right-hand side is the magnitude of the cross product of the partial derivatives of x(s, t), and is known as the surface element. for example, if we want to find the surface area of some general scalar function, say z=f\,(x,y), we have : a = \iint_s \,\mathrm ds = \iint_t \left\|{\partial \mathbf{r} \over \partial x}\times {\partial \mathbf{r} \over \partial y}\right\| \mathrm dx\, \mathrm dy where \mathbf{r}=(x, y, z)=(x, y, f(x,y)). so that {\partial \mathbf{r} \over \partial x}=(1, 0, f_x(x,y)), and {\partial \mathbf{r} \over \partial y}=(0, 1, f_y(x,y)). so, :\begin{align} a &{} = \iint_t \left\|\left(1, 0, {\partial f \over \partial x}\right)\times \left(0, 1, {\partial f \over \partial y}\right)\right\| \mathrm dx\, \mathrm dy \\ &{} = \iint_t \left\|\left(-{\partial f \over \partial x}, -{\partial f \over \partial y}, 1\right)\right\| \mathrm dx\, \mathrm dy \\ &{} = \iint_t \sqrt{\left({\partial f \over \partial x}\right)^2+\left({\partial f \over \partial y}\right)^2+1}\, \, \mathrm dx\, \mathrm dy \end{align} which is the familiar formula we get for the surface area of a general functional shape. one can recognize the 
Area Multivariable_calculus Surfaces