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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.
In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero (dα = 0), and an exact form is a differential form, α, that is the exterior derivative of another differential form β. Thus, an exact form is in the image of d, and a closed form is in the kernel of d.
Differential geometry is also indispensable in the study of gravitational lensing and black holes. Differential forms are used in the study of electromagnetism. Differential geometry has applications to both Lagrangian mechanics and Hamiltonian mechanics. Symplectic manifolds in particular can be used to study Hamiltonian systems.
That is, df is the unique 1-form such that for every smooth vector field X, df (X) = d X f , where d X f is the directional derivative of f in the direction of X. The exterior product of differential forms (denoted with the same symbol ∧) is defined as their pointwise exterior product.
Pages in category "Differential forms" The following 22 pages are in this category, out of 22 total. This list may not reflect recent changes. ...
The wedge product of complex differential forms is defined in the same way as with real forms. Let p and q be a pair of non-negative integers ≤ n. The space Ω p,q of (p, q)-forms is defined by taking linear combinations of the wedge products of p elements from Ω 1,0 and q elements from Ω 0,1. Symbolically,
Discrete differential calculus is the study of the definition, properties, and applications of the difference quotient of a function. ... (differential) forms, ...
Another generalization, due to Albert Nijenhuis, allows one to define the Lie derivative of a differential form along any section of the bundle Ω k (M, TM) of differential forms with values in the tangent bundle. If K ∈ Ω k (M, TM) and α is a differential p-form, then it is possible to define the interior product i K α of K and α. The ...