Editing Integral (section)

=== Line integrals and surface integrals ===
{{Main|Line integral|Surface integral}}
[[File:Line-Integral.gif|right|thumb|A line integral sums together elements along a curve.]]
The concept of an integral can be extended to more general domains of integration, such as curved lines and surfaces inside higher-dimensional spaces. Such integrals are known as line integrals and surface integrals respectively. These have important applications in physics, as when dealing with [[vector field]]s.

A ''line integral'' (sometimes called a ''path integral'') is an integral where the [[Function (mathematics)|function]] to be integrated is evaluated along a [[curve]].<ref>{{Harvnb|Anton|Bivens|Davis|2016|p=980}}.</ref> Various different line integrals are in use. In the case of a closed curve it is also called a ''contour integral''.

The function to be integrated may be a [[scalar field]] or a [[vector field]]. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly [[arc length]] or, for a vector field, the [[Inner product space|scalar product]] of the vector field with a [[Differential (infinitesimal)|differential]] vector in the curve).<ref>{{Harvnb|Anton|Bivens|Davis|2016|p=981}}.</ref> This weighting distinguishes the line integral from simpler integrals defined on [[Interval (mathematics)|intervals]]. Many simple formulas in physics have natural continuous analogs in terms of line integrals; for example, the fact that [[Mechanical work|work]] is equal to [[force]], {{math|'''F'''}}, multiplied by displacement, {{math|'''s'''}}, may be expressed (in terms of vector quantities) as:<ref>{{Harvnb|Anton|Bivens|Davis|2016|p=697}}.</ref>

: <math>W=\mathbf F\cdot\mathbf s.</math>

For an object moving along a path {{mvar|''C''}} in a [[vector field]] {{math|'''F'''}} such as an [[electric field]] or [[gravitational field]], the total work done by the field on the object is obtained by summing up the differential work done in moving from {{math|'''s'''}} to {{math|'''s''' + ''d'''''s'''}}. This gives the line integral<ref>{{Harvnb|Anton|Bivens|Davis|2016|p=991}}.</ref>

: <math>W=\int_C \mathbf F\cdot d\mathbf s.</math>
[[File:Surface_integral_illustration.svg|right|thumb|The definition of surface integral relies on splitting the surface into small surface elements.]]
A ''surface integral'' generalizes double integrals to integration over a [[Surface (mathematics)|surface]] (which may be a curved set in [[space]]); it can be thought of as the [[Multiple integral|double integral]] analog of the [[line integral]]. The function to be integrated may be a [[scalar field]] or a [[vector field]]. The value of the surface integral is the sum of the field at all points on the surface. This can be achieved by splitting the surface into surface elements, which provide the partitioning for Riemann sums.<ref>{{Harvnb|Anton|Bivens|Davis|2016|p=1014}}.</ref>

For an example of applications of surface integrals, consider a vector field {{math|'''v'''}} on a surface {{math|''S''}}; that is, for each point {{math|''x''}} in {{math|''S''}}, {{math|'''v'''(''x'')}} is a vector. Imagine that a fluid flows through {{math|''S''}}, such that {{math|'''v'''(''x'')}} determines the velocity of the fluid at {{mvar|x}}. The [[flux]] is defined as the quantity of fluid flowing through {{math|''S''}} in unit amount of time. To find the flux, one need to take the [[dot product]] of {{math|'''v'''}} with the unit [[Normal (geometry)|surface normal]] to {{math|''S''}} at each point, which will give a scalar field, which is integrated over the surface:<ref>{{Harvnb|Anton|Bivens|Davis|2016|p=1024}}.</ref>

: <math>\int_S {\mathbf v}\cdot \,d{\mathbf S}.</math>

The fluid flux in this example may be from a physical fluid such as water or air, or from electrical or magnetic flux. Thus surface integrals have applications in physics, particularly with the [[classical theory]] of [[electromagnetism]].