# 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.

## 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 ωAk(Rn)

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

It is said that I satisfy を. k-1 next differential calculus form ηAk-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, Hk(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; Zk(M)

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

Whole k next differential calculus form, Bk(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.

## 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