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The most basic non-trivial differential one-form is the "change in angle" form . This is defined as the derivative of the angle "function" θ ( x , y ) {\\displaystyle \\theta (x,y)} (which is only defined up to an additive constant), which can be explicitly defined in terms of the atan2 function.
Differential forms are part of the field of differential geometry, influenced by linear algebra. Although the notion of a differential is quite old, the initial attempt at an algebraic organization of differential forms is usually credited to Élie Cartan with reference to his 1899 paper. [1]
Differential geometry is a ... geometric calculus both extrinsic and intrinsic geometry of a manifold can be characterized by a single bivector-valued one-form ...
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.
A one-form () corresponds to the unique vector field ♯ such that for all (), we have: α ( X ) = g ( α ♯ , X ) . {\displaystyle \alpha (X)=g(\alpha ^{\sharp },X).} These mappings extend via multilinearity to mappings from k {\displaystyle k} -vector fields to k {\displaystyle k} -forms and k {\displaystyle k} -forms to k {\displaystyle k ...
On a Riemann surface the Poincaré lemma states that every closed 1-form or 2-form is locally exact. [2] Thus if ω is a smooth 1-form with dω = 0 then in some open neighbourhood of a given point there is a smooth function f such that ω = df in that neighbourhood; and for any smooth 2-form Ω there is a smooth 1-form ω defined in some open neighbourhood of a given point such that Ω = dω ...
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 ...
One-form (differential geometry) P. Poincaré lemma; Positive form; S. Solder form; Sum of residues formula; V. Vector-valued differential form; Volume form