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,

${\displaystyle \mathrm {d} \omega =0\,}$ 

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,

${\displaystyle \omega =\mathrm {d} \eta \,}$ 

When k-1 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

${\displaystyle \mathrm {d} \circ \mathrm {d} =0\,}$ 

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 Rⁿ (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(Rⁿ)

${\displaystyle \mathrm {d} \omega =0\,}$ 

It is said that I satisfy を. k-1 next differential calculus form η ∈ A^k-1(Rⁿ) exists then and,

${\displaystyle \omega =\mathrm {d} \eta \,}$ 

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.

${\displaystyle H^{k}(\mathbb {R} ^{n})=\left\{{\begin{matrix}\mathbb {R} &(k=0)\\0&(k>0)\end{matrix}}\right.}$ 

But, for manifold M, H^k(M) is a quotient vector space

${\displaystyle H^{k}(M)=Z^{k}(M)/B^{k}(M)\,}$ 

Is ド Lahm cohomology group of the で defined k next; Z^k(M)

${\displaystyle Z^{k}(M)=\mathrm {ker} \,\mathrm {d} \cap A^{k}(M)\,}$ 

Whole k next differential calculus form, B^k(M) of the shut form defined で

${\displaystyle B^{k}(M)=\mathrm {im} \,\mathrm {d} \cap A^{k}(M)\,}$ 

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(Rⁿ)) becomes the complete form (the cause of B ^k(Rⁿ)).

Expansion

Generally next consists about manifold M which is a possible shrinkage.

${\displaystyle H^{k}(M)=0\quad (k>0).}$ 

Specific example

The primary differential calculus form that is defined, for example, on R²

${\displaystyle \omega _{1}=xy^{2}\mathrm {d} x+x^{2}y\mathrm {d} y\,}$ 

Oh, when I think about outside differential calculus

${\displaystyle \mathrm {d} \omega _{1}=2xy\,\mathrm {d} y\wedge \mathrm {d} x+2xy\,\mathrm {d} x\wedge \mathrm {d} y=0}$ 

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 R²

${\displaystyle \eta _{1}={\frac {1}{2}}x^{2}y^{2}\,}$ 

It is,

${\displaystyle d\eta _{1}=xy^{2}dx+x^{2}y\,dy=\omega _{1}\,}$ 

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 R²∖(0, 0) except the origin by R²

${\displaystyle \omega _{2}={\frac {-y}{x^{2}+y^{2}}}\,\mathrm {d} x+{\frac {x}{x^{2}+y^{2}}}\,\mathrm {d} y}$ 

Oh, when I think about outside differential calculus

${\displaystyle d\omega _{2}=0\,}$ 

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 R²∖{x = 0} except the x-axis by R²

${\displaystyle \eta _{2}=\arctan {\frac {y}{x}}}$ 

It is,

${\displaystyle \mathrm {d} \eta _{2}={\frac {-y}{x^{2}+y^{2}}}\,\mathrm {d} x+{\frac {x}{x^{2}+y^{2}}}\,\mathrm {d} y}$ 

I accord with ω ₂, but, であり, η ₂ is not defined in R²∖(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 three-dimensional vector field F defined in the whole R³

${\displaystyle \operatorname {rot} \mathbf {F} =\mathbf {0}}$ 

If I satisfy を,

${\displaystyle \mathbf {F} =\operatorname {grad} \psi}$ 

Scalar potential ψ on R³ satisfying の relations exists. In this case F = (F₁, F₂, F₃) is the first differential calculus form

${\displaystyle \omega =F_{1}\mathrm {d} x+F_{2}\mathrm {d} y+F_{3}\mathrm {d} z\,}$ 

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 R³.

Existence of the vector potential

Similarly, emission div in three-dimensional vector field G defined in the whole R³

${\displaystyle \operatorname {div} \mathbf {G} = 0}$ 

If I satisfy を,

${\displaystyle \mathbf {G} =\operatorname {rot} \mathbf {A}}$ 

Vector potential A on R³ satisfying の relations exists. In this case G = (G₁, G₂, G₃) is the second differential calculus form

${\displaystyle \omega =G_{1}\mathrm {d} y\wedge \mathrm {d} z+G_{2}\mathrm {d} z\wedge \mathrm {d} x+G_{3}\mathrm {d} x\wedge \mathrm {d} y\,}$ 

Make に correspondence; a differential calculus form primary as for A = (A₁, A₂, A₃)

${\displaystyle \eta =A_{1}\mathrm {d} x+A_{2}\mathrm {d} y+A_{3}\mathrm {d} z\,}$ 

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

• Differential calculus form
• ド Lahm cohomology
• Henri Poincare

References

• Bott, Raoul; Tu, Loring W. (1995). Differential Forms in Algebraic Topology. Springer. ISBN 978-0387906133. 

• • 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 978-4,431,707,073。

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