Editing Integral (section)

=== Integrals of differential forms ===
{{Main|Differential form}}
{{See also|Volume form|Density on a manifold}}
A [[differential form]] is a mathematical concept in the fields of [[multivariable calculus]], [[differential topology]], and [[tensor]]s. Differential forms are organized by degree. For example, a one-form is a weighted sum of the differentials of the coordinates, such as:

: <math>E(x,y,z)\,dx + F(x,y,z)\,dy + G(x,y,z)\, dz</math>

where ''E'', ''F'', ''G'' are functions in three dimensions. A differential one-form can be integrated over an oriented path, and the resulting integral is just another way of writing a line integral. Here the basic differentials ''dx'', ''dy'', ''dz'' measure infinitesimal oriented lengths parallel to the three coordinate axes.

A differential two-form is a sum of the form

: <math>G(x,y,z) \, dx\wedge dy + E(x,y,z) \, dy\wedge dz + F(x,y,z) \, dz\wedge dx.</math>

Here the basic two-forms <math>dx\wedge dy, dz\wedge dx, dy\wedge dz</math> measure oriented areas parallel to the coordinate two-planes. The symbol <math>\wedge</math> denotes the [[wedge product]], which is similar to the [[cross product]] in the sense that the wedge product of two forms representing oriented lengths represents an oriented area. A two-form can be integrated over an oriented surface, and the resulting integral is equivalent to the surface integral giving the flux of <math>E\mathbf i+F\mathbf j+G\mathbf k</math>.

Unlike the cross product, and the three-dimensional vector calculus, the wedge product and the calculus of differential forms makes sense in arbitrary dimension and on more general manifolds (curves, surfaces, and their higher-dimensional analogs). The [[exterior derivative]] plays the role of the [[gradient]] and [[Curl (mathematics)|curl]] of vector calculus, and [[Generalized Stokes theorem|Stokes' theorem]] simultaneously generalizes the three theorems of vector calculus: the [[divergence theorem]], [[Green's theorem]], and the [[Kelvin-Stokes theorem]].