Integral
Integrals of differential forms

A differential form is a mathematical concept in the fields of multivariable calculus, differential topology, and tensors. Differential forms are organized by degree. For example, a one-form is a weighted sum of the differentials of the coordinates, such as:

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

Here the basic two-forms d x ∧ d y, d z ∧ d x, d y ∧ d z {\displaystyle dx\wedge dy,dz\wedge dx,dy\wedge dz} measure oriented areas parallel to the coordinate two-planes. The symbol ∧ {\displaystyle \wedge } 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 E i + F j + G k {\displaystyle E\mathbf {i} +F\mathbf {j} +G\mathbf {k} }.

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 of vector calculus, and Stokes' theorem simultaneously generalizes the three theorems of vector calculus: the divergence theorem, Green's theorem, and the Kelvin-Stokes theorem.
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