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Concept
Differential form
Summary
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, 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.
For instance, the expression f(x) dx is an example of a 1-form, and can be integrated over an interval [a, b] contained in the domain of f:
Similarly, the expression f(x, y, z) dx ∧ dy + g(x, y, z) dz ∧ dx + h(x, y, z) dy ∧ dz is a 2-form that can be integrated over a surface S:
The symbol ∧ denotes the exterior product, sometimes called the wedge product, of two differential forms. Likewise, a 3-form f(x, y, z) dx ∧ dy ∧ dz represents a volume element that can be integrated over a region of space. In general, a k-form is an object that may be integrated over a k-dimensional manifold, and is homogeneous of degree k in the coordinate differentials
On an n-dimensional manifold, the top-dimensional form (n-form) is called a volume form.
The differential forms form an alternating algebra. This implies that and This alternating property reflects the orientation of the domain of integration.
The exterior derivative is an operation on differential forms that, given a k-form , produces a (k+1)-form This operation extends the differential of a function (a function can be considered as a 0-form, and its differential is ) This allows expressing the fundamental theorem of calculus, the divergence theorem, Green's theorem, and Stokes' theorem as special cases of a single general result, the generalized Stokes theorem.
Differential 1-forms are naturally dual to vector fields on a differentiable 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.
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