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Differential form - Integration of forms | | Differential form - Integration of forms: Encyclopedia II - Differential form - Integration of forms | | Differential forms of degree k are integrated over k dimensional chains. If k = 0, this is just evaluation of functions at points. Other values of k = 1, 2, 3, ... correspond to line integrals, surface integrals, volume integrals etc.
Let
be a differential form and S a set for which we wish to integrate over, where S has the parameterization
for u in the parameter domain D. Then [Rudin, 1976] defines the integral of the differential form over S as
where
< ...
See also:Differential form, Differential form - Gentle introduction, Differential form - Properties of the wedge product, Differential form - Formal definition, Differential form - Integration of forms, Differential form - Operations on forms | | | Differential form, Differential form - Formal definition, Differential form - Gentle introduction, Differential form - Integration of forms, Differential form - Operations on forms, Differential form - Properties of the wedge product, complex form | | |
| | | Differential form: Encyclopedia II - Differential form - Integration of forms
Differential form - Integration of forms
Differential forms of degree k are integrated over k dimensional chains. If k = 0, this is just evaluation of functions at points. Other values of k = 1, 2, 3, ... correspond to line integrals, surface integrals, volume integrals etc.
Let
be a differential form and S a set for which we wish to integrate over, where S has the parameterization
for u in the parameter domain D. Then [Rudin, 1976] defines the integral of the differential form over S as
where
is the determinant of the Jacobian.
See also Stokes' theorem.
Other related archives1-forms, Elie Cartan, Jacobian, Lie derivative, Riemann sums, Stokes' theorem, anticommutativity, antisymmetric, chains, complex form, cotangent bundle, covariant, covariant vectors, cover, de Rham cohomology, differential, differential geometry, differential topology, dimensional, dual space, exterior algebra, exterior derivative, exterior derivatives, exterior power, function, homology, integrate, manifold, manifolds, multilinear map, multivariate calculus, open set, smooth function, tangent space, tensor, tensors, vector space, wedge product, wedge products
 Adapted from the Wikipedia article "Integration of forms", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
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