In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to defining integrands over curves, surfaces, volumes, and higher dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.
and similarly the expression: f(x,y,z) dx∧dy + g(x,y,z) dx∧dz + h(x,y,z) dy∧dz is a 2-form that has a surface integral over an oriented surface S:
Likewise, a 3-form f(x, y, z) dx∧dy∧dz represents something that can be integrated over a region of space.
The algebra of differential forms is organized in a way that naturally reflects the orientation of the domain of integration. There is an operation d on differential forms known as the exterior derivative that, when acting on a k-form produces a (k+1)-form. This operation extends the differential of a function, and the divergence and the curl of a vector field in an appropriate sense that makes the fundamental theorem of calculus, the divergence theorem, Green's theorem, and Stokes' theorem special cases of the same general result, known in this context also as the general Stokes' theorem. In a deeper way, this theorem relates the topology of the domain of integration to the structure of the differential forms themselves; the precise connection is known as De Rham's theorem.
The general setting for the study of differential forms is on a differentiable manifold. Differential 1-forms are naturally dual to vector fields on a manifold, and the pairing between vector fields and 1-forms is extended to arbitrary differential forms by the interior product. The algebra of differential forms along with the exterior derivative defined on it is preserved by the pullback under smooth functions between two manifolds. This feature allows geometrically invariant information to be moved from one space to another via the pullback, provided the information is expressed in terms of differential forms. As a particular example, the change of variables formula for integration becomes a simple statement that an integral is preserved under pullback.
Other articles related to "differential form, differentials, differential, forms, form":
... Consider the following differential form of a constraint equation where cij, ci are the coefficients of the differentials dqj and dt for the ith constraint ... If the differential form is integrable, i.e ... some nonholonomic constraints can be expressed using the differential form ...
... It is often easier to work in differential form and then convert back to normal derivatives ... Differential identities scalar involving matrix Condition Expression Result (numerator layout) Differential identities matrix Condition Expression Result (numerator layout) A ...
... The two forms of Gauss's law for gravity are mathematically equivalent ... the divergence theorem to the integral form of Gauss's law for gravity, which becomes which can be rewritten This has to hold simultaneously for every possible volume V ... Hence we arrive at which is the differential form of Gauss's law for gravity ...
... Numerous minimality results for complex analytic manifolds are based on the Wirtinger inequality for 2-forms ... A succinct proof may be found in Herbert Federer's classic text Geometric Measure Theory ...
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