In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. A differential form of degree k, or (differential) k-form, on a smooth manifold M is a smooth section of the kth exterior power of the cotangent bundle of M. The set of all k-forms on M is a vector space commonly denoted Ωk(M).
A differential 0-form is by definition a smooth function on M. A differential 1-form is an object dual to a vector field on M.
Differential forms can be multiplied together using an operation called the wedge product. There is also a differential operator on differential forms called the exterior derivative. The wedge product of a k-form and an l-form is a (k+l)-form, and the exterior derivative of a k-form is a (k+1)-form. In particular, the exterior derivative of a 0-form (which is a function on M) is its differential (which is a 1-form on M).
The modern notion of differential forms was pioneered by Élie Cartan, and has many applications, especially in geometry, topology and physics.
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Lorenzo
Fri, 24 Jul 2009 19:44:00 GM
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