Surface integral

Integration over a non-flat region in 3D space The definition of the surface integral relies on splitting the surface into small surface elements. In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one may integrate over this surface a scalar field (that is, a function of position which returns a scalar as a value), or a vector field (that is, a function which returns a vector as value). If a region R is not flat, then it is called a surface as shown in the illustration. Surface integrals have applications in physics, particularly in the classical theories of electromagnetism and fluid mechanics. 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 Assume that f is a scalar, vector, or tensor field defined on a surface S. To find an explicit formula for the surface integral of f over S, we need to parameterize S by defining a system of curvilinear coordinates on S, like