differential form

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differential form

[‚dif·ə′ren·chəl ′fȯrm]
(mathematics)
A homogeneous polynomial in differentials.
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where B and F are a (n - 2)-form and a 2-form respectively on M and which takes values in the Lie algebra g of G.
[OMEGA] = ([[OMEGA].sup.i.sub.j]) is called curvature 2-form or the Riemannian curvature tensor associated with the connection [theta] and it is written as
For example, with two 1-forms e and h in 3-D space, their wedge product e [conjunction] h is a 2-form. And this corresponds to the fact that the cross product of two vector fields E(r) and H(r) gives rise to a new vector field S(r) = E x H.
G is also equivalent to a pair ([gamma], [psi]) where [gamma] is a Riemannian metric and [psi] is a 2-form on M.
The fundamental 2-form [PHI] on an almost contact metric manifold [M.sup.2n+1] is defined by [PHI](X, Y) = g(X, [phi]Y) for any vector fields X and Y on [M.sup.2n+1].
We recall that multiplicative of a 2-form w is defined by
The samples show typical diffraction peak of monoclinic 2-form crystals, indicating that during cold-temperature stretching no crystal transformation occurrence.
For instance, if [[A.sub.12]d[x.sub.1] [and] d[x.sub.2]] = {[A.sub.12]d[x.sub.1] [and] d[x.sub.2], -[A.sub.12]d[x.sub.2] [and] d[x.sub.1]} [member of] [E.sup.2], then [A.sub.12]d[x.sub.1] [and] d[x.sub.2] is chosen as a representative of the considered 2-form. By the constructions described above we obtain a sequence of vector spaces [E.sup.0] : = [K.sup.*], [E.sup.1] := E, [E.sup.2], [E.sup.3], [E.sup.p], ...
So, we define the electromagnetic strength F 2-form as:
The 2-form [PHI] on [M.sup.2n+1] defined by [PHI] (X, Y) = g (X,(Y) is called the fundamental 2-form of the almost contact metric manifold [M.sup.2n+1].