Lemma of Poincare
One of the theorems in the phase geometry algebraic in mathematics with the lemma (ぽあんかれのほだい British: Poincarélemma) of Poincare. In Euclidean space, I insist on the differential calculus formality that is a shut form becoming the complete form.
Table of contents
Summary
Introduction
Outside differential calculus dω about differential calculus form ω of the k next in the manifold,
となる ω is called a shut form (closed form). Or it is the same thing, but the nuclear cause of d is called a shut form. In addition, for k next differential calculus form ω,
When k1 next differential calculus form η satisfying を exists, it is said that ω is a complete form (exact form). Or it is the same thing, but the cause of the image of d is called a complete form. In addition, η is often called potential.
Outside differential property
It is always managed that a more complete form is a shut form, but a shut form is different by the geometric property of the manifold whether it is in a perfection form.
The lemma of Poincare insists on next:
 In "Euclidean space R^{n} (manifold M which is generally a possible shrinkage), any shut form is a complete form"
Claim of the theorem
k > Assume it 0; k next differential calculus form ω ∈ A^{k}(R^{n})
It is said that I satisfy を. k1 next differential calculus form η ∈ A^{k1}(R^{n}) exists then and,
But, it is managed.
Expression by the ド Lahm cohomology
I can express the lemma of Poincare as follows if I use a concept of the ド Lahm cohomology.
But, for manifold M, H^{k}(M) is a quotient vector space
Is ド Lahm cohomology group of the で defined k next; Z^{k}(M)
Whole k next differential calculus form, B^{k}(M) of the shut form defined で
で is the whole k next differential calculus form of the defined complete form.
If is merely df(x) ≡ 0, in the case of k = 0, speak that f becomes the constant function; k > It becomes a lemma of Poincare whom 0 cases mentioned above and the equivalent expression. In other words, I express that a shut form (the cause of Z ^{k}(R^{n})) becomes the complete form (the cause of B ^{k}(R^{n})).
Expansion
Generally next consists about manifold M which is a possible shrinkage.
Specific example
The primary differential calculus form that is defined, for example, on R^{2}
Oh, when I think about outside differential calculus
It is the neighbor, a shut form. Therefore, it becomes the form more complete than a lemma of Poincare. Fact, the zeroth differential calculus form on R^{2}
It is,
But, ω _{1} is a complete form because it is managed.
On the other hand, the primary differential calculus form that is defined in domain R^{2}∖(0, 0) except the origin by R^{2}
Oh, when I think about outside differential calculus
But, ω _{2} is a shut form because it is managed. However, the domain to think about does not meet the condition of the lemma of Poincare, and it is not guaranteed that ω _{2} is a complete form. The zeroth differential calculus form that is defined with domain R^{2}∖{x = 0} except the xaxis by R^{2}
It is,
I accord with ω _{2}, but, であり, η _{2} is not defined in R^{2}∖(0, 0) locally.
with the vector analysis of relationships
The existence condition of scalar potential and the vector potential in the vector analysis of the lemma of Poincare when is special, is equivalent.
Existence of the scalar potential
Rotary rot in threedimensional vector field F defined in the whole R^{3}
If I satisfy を,
Scalar potential ψ on R^{3} satisfying の relations exists. In this case F = (F_{1}, F_{2}, F_{3}) is the first differential calculus form
I make に correspondence, and ψ supports the zeroth differential calculus form η. In addition, I correspond to a primary differential calculus form, and the action of rotary rot is equivalent to differential calculus. In addition, as a condition of the domains of the vector fields, I can take the domain that is a single connection as well as the whole R^{3}.
Existence of the vector potential
Similarly, emission div in threedimensional vector field G defined in the whole R^{3}
If I satisfy を,
Vector potential A on R^{3} satisfying の relations exists. In this case G = (G_{1}, G_{2}, G_{3}) is the second differential calculus form
Make に correspondence; a differential calculus form primary as for A = (A_{1}, A_{2}, A_{3})
I make に correspondence. In addition, I correspond to the second differential calculus form, and the action of emission div is equivalent to differential calculus.
Allied item
References
 Bott, Raoul; Tu, Loring W. (1995). Differential Forms in Algebraic Topology. Springer. ISBN 9780387906133.

 Bott, Raoul, Tu, Loring W. "Differential calculus form and algebra topology" Mimura, 護 (translation), シュプリンガー fair Lark Tokyo, 1996. ISBN 9784,431,707,073。
This article is taken from the Japanese Wikipedia Lemma of Poincare
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