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An example of a 1-dimensional manifold is an interval [a, b], and intervals can be given an orientation: they are positively oriented if a < b, and negatively oriented otherwise. If a < b then the integral of the differential 1-form f(x) dx over the interval [a, b] (with its natural positive orientation) is
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.
(For instance, when n = 3, i.e. in three-dimensional space, the 2-form ω V is locally the scalar triple product with V.) The integral of ω V over a hypersurface is the flux of V over that hypersurface. The exterior derivative of this (n − 1)-form is the n-form
A local 1-form on M is a contact form if the restriction of its exterior derivative to H is a non-degenerate two-form and thus induces a symplectic structure on H p at each point. If the distribution H can be defined by a global one-form then this form is contact if and only if the top-dimensional form
This is a (dualized) (1 + 1)-dimensional case, for a 1-form (dualized because it is a statement about vector fields). This special case is often just referred to as Stokes' theorem in many introductory university vector calculus courses and is used in physics and engineering.
In 3 dimensions, a differential 0-form is a real-valued function (,,); a differential 1-form is the following expression, where the coefficients are functions: + +; a differential 2-form is the formal sum, again with function coefficients: + +; and a differential 3-form is defined by a single term with one function as coefficient: .
Given a complex manifold M, a differential of the first kind ω is therefore the same thing as a 1-form that is everywhere holomorphic; on an algebraic variety V that is non-singular it would be a global section of the coherent sheaf Ω 1 of Kähler differentials. In either case the definition has its origins in the theory of abelian integrals.
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 ...