What is Surface integral?
{{jargon|date=January 2024}} In [[mathematics]], a '''surface integral''' is a definite [[integral]] taken over a [[surface]] (which may be a [[curve]] [[Set (mathematics)|set]] in [[Space (physics)|space]]). Just as a line integral allows one to integrate over an arbitrary curve (of one [[dimension]]), a surface integral can be thought of as a [[multiple integral|double integral]] integrating over a two-dimensional surface. Given a surface, one may integrate over its scalar fields (that is, [[function (mathematics)|function]]s which return [[number]]s as values), or its [[vector field]]s (that is, functions which return [[Vector|vectors]] as values). Surface integrals have applications in [[physics]], particularly in the classical theories of [[electromagnetism]] and [[fluid mechanics]]. [[Image:Surface integral illustration.svg|right|thumb|The definition of surface integral relies on splitting the surface into small surface elements.]] [[Image:Surface integral1.svg|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 == Consider a surface ''S'' on which a scalar field ''f'' is defined. If one thinks of ''S'' as made of some [[material]], and for each '''x''' in ''S'' the number ''f''('''x''') is the [[density]] of material at '''x''', then the surface integral of ''f'' over ''S'' is the [[mass]] per unit thickness of ''S''. (This is only true if the surface is an [[Infinitesimal|infinitesimally]] thin shell.) One approach to calculating the surface integral is then to split the surface in many very small pieces, assume that the density is approximately [[constant]] on each piece, find the mass per unit thickness of each piece (by multiplying the density of the piece by its area), and then sum up the resulting numbers to find the total mass per unit thickness of ''S''. To find an explicit formula for the surface integral, mathematicians [[Coordinate syste