Poincaré lemma

The Poincaré lemma is a mathematical theorem and was named after the French mathematician Henri Poincaré .

Exact and closed differential forms

• A differential form of degree is called closed if applies. The outer derivation denotes . ${\displaystyle \omega }$ ${\displaystyle k}$ ${\displaystyle \mathrm {d} \omega =0}$ ${\displaystyle \mathrm {d} }$
• A differential form of degree is called exact if there is a differential form such that it holds. The form is called a potential form of ${\displaystyle \omega }$ ${\displaystyle k}$ ${\displaystyle (k-1)}$ ${\displaystyle \nu }$ ${\displaystyle \omega =\mathrm {d} \nu }$ ${\displaystyle \nu }$ ${\displaystyle \omega .}$

The potential form is not clearly defined, but only "except for re-calibration" (see below).

Because of this , every exact differential form is also closed. The Poincaré lemma specifies conditions under which the opposite statement also applies. In the proof, there is also a generalization of the lemma: An exact part can be split off from every differential form “by construction”. ${\displaystyle \mathrm {d} \circ \mathrm {d} \equiv 0}$

statement

The Poincaré lemma says that every closed differential form defined on a star-shaped open set is exact. ${\displaystyle U\subseteq \mathbb {R} ^{d}}$

The statement can also be formulated more abstractly as follows: For a star-shaped open set , the -th De Rham cohomology vanishes for all : ${\displaystyle U\subseteq \mathbb {R} ^{d}}$ ${\displaystyle k}$ ${\displaystyle k>0}$

${\displaystyle \mathrm {H} _{\mathrm {dR} }^{k}(U)=0}$

In the three-dimensional special case, the Poincaré lemma, translated into the language of vector analysis , says that an eddy-free vector field defined on a simply connected area as a gradient of a potential field ( ), a source-free vector field on a convex area through rotation of a vector potential ( ), and a scalar field density ("source density") can be represented as the divergence of a vector field ( ). ${\displaystyle \Phi (\mathbf {r} )}$ ${\displaystyle k=1}$ ${\displaystyle {\vec {A}}(\mathbf {r} ,t)}$ ${\displaystyle k=2}$ ${\displaystyle k=3}$

Proof (constructive)

Be the point around which is star-shaped. The Poincaré Lemma gives an explicit form, and although having the formula: any form unity can not necessary provided that a form assign, from which the desired potential shape results in unity: This associated shape can by itself define the following figure: ${\displaystyle x_{0}\in U}$ ${\displaystyle U\subset \mathbb {R} ^{n}}$ ${\displaystyle (k-1)}$ ${\displaystyle k}$ ${\displaystyle \textstyle \omega ^{k}=\sum \omega _{I}{\rm {d}}x_{I}}$ ${\displaystyle (k-1)}$ ${\displaystyle P^{k-1}(\omega ^{k})}$

${\displaystyle P^{k-1}(\omega ^{k})(x)\,:=\,\sum _{i_{1}<\cdots <i_{k}}\sum _{\alpha =1}^{k}(-1)^{\alpha -1}{\Big (}\int _{0}^{1}t^{k-1}\omega _{i_{1}\cdots i_{k}}(x_{0}+tx)dt{\Big )}x^{i_{\alpha }}{\rm {d}}x^{i_{1}}\wedge \cdots \wedge {\widehat {{\rm {d}}x^{i_{\alpha }}}}\wedge \cdots \wedge {\rm {d}}x^{i_{k}}}$.

(The roof symbol in the -th column on the right-hand side means that the corresponding differential is omitted.) ${\displaystyle i_{\alpha }}$

Now one shows directly that the following identity applies: what formally corresponds to the product rule of differentiation and splits the properties represented by into two parts, of which the second has the property sought. ${\displaystyle \omega ^{k}\equiv \mathrm {P} ^{k}({\rm {d}}\omega ^{k})+{\rm {d}}{\mathrm {P} ^{k-1}(}\omega ^{k}\mathrm {)} \,,}$ ${\displaystyle \omega ^{k}}$

Because of the prerequisite and because of this, this initially applies without restricting the generality even without the foremost of the right-hand side, namely because the requirement only considers the form at the zero point, so that, as with the total differential, a function from up to so-called Calibration transformations (see below) can also be deduced. ${\displaystyle {\rm {d}}\omega ^{k}\equiv 0}$ ${\displaystyle \mathrm {d} \circ \mathrm {d} =0}$ ${\displaystyle 0\equiv {\mathrm {d} }P^{k}({\mathrm {d} {\omega }^{k}}\to 0).}$ ${\displaystyle \mathrm {d} }$ ${\displaystyle \mathrm {d} \omega ^{k}\to 0}$ ${\displaystyle \mathrm {d} P^{k}}$ ${\displaystyle \mathrm {d} P^{k}=0}$ ${\displaystyle \mathrm {P} ^{k}=0}$

This leaves only the last term of the above identity, and the required statement follows: with ${\displaystyle \omega ^{k}\equiv \mathrm {d} \eta ^{k-1},}$ ${\displaystyle \eta ^{k-1}\,:=\,\mathrm {P} ^{k-1}(\omega ^{k})\,.}$

The given identity also generalizes Poincaré's lemma by breaking down any differential form into an inexact (“anholonomic”) and an exact (“holonomic”) part (the bracketed names correspond to the so-called constraining forces in analytical mechanics ). At the same time, it corresponds to the decomposition of any vector field into a vortex part and a source part. ${\displaystyle \omega }$

In the language of homological algebra , a contracting homotopy , e.g. B. contracted to the central point of the star-shaped area considered here . ${\displaystyle P}$

Re-calibration

What is so defined is not the only form whose external differential is. All others differ from one another by the differential of a -form: If and are two such -forms, then there exists a -form such that it holds. ${\displaystyle \eta ^{k-1}}$ ${\displaystyle (k-1)}$ ${\displaystyle {\omega }^{k}}$ ${\displaystyle (k-2)}$ ${\displaystyle \eta _{2}^{k-1}}$ ${\displaystyle \eta _{1}^{k-1}}$ ${\displaystyle (k-1)}$ ${\displaystyle (k-2)}$ ${\displaystyle \xi ^{k-2}}$ ${\displaystyle \eta _{2}^{k-1}=\eta _{1}^{k-1}+\mathrm {d} \xi ^{k-2}}$

The addition is also known as calibration transformation or re-calibration of . ${\displaystyle +\,\mathrm {d} \xi ^{k-2}}$ ${\displaystyle \eta _{1}^{k-1}}$

Application in electrodynamics

The case of a magnetic field generated by a stationary current is known from electrodynamics , with the so-called vector potential . This case corresponds , the star-shaped area being the . The vector of the current density is and corresponds to the current shape. The same applies to the magnetic field : it corresponds to the magnetic flux shape and can be derived from the vector potential:, or . The vector potential corresponds to the potential form. The closeness of the magnetic flux form corresponds to the absence of sources of the magnetic field ${\displaystyle {\vec {A}}(\mathbf {r} )\,.}$ ${\displaystyle k=2}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle {\vec {j}}}$ ${\displaystyle \mathbf {I} :=j_{1}(x,y,z){\rm {d}}x_{2}\wedge {\rm {d}}x_{3}+j_{2}(x,y,z){\rm {d}}x_{3}\wedge {\rm {d}}x_{1}+j_{3}(x,y,z){\rm {d}}x_{1}\wedge {\rm {d}}x_{2}\,.}$ ${\displaystyle {\vec {B}}}$ ${\displaystyle \Phi _{B}:=B_{1}{\rm {d}}x_{2}\wedge {\rm {d}}x_{3}+\dots }$ ${\displaystyle \textstyle {\vec {B}}=\operatorname {rot} {\vec {A}}=\left({\tfrac {\partial A_{3}}{\partial x_{2}}}-{\tfrac {\partial A_{2}}{\partial x_{3}}},{\tfrac {\partial A_{1}}{\partial x_{3}}}-{\tfrac {\partial A_{3}}{\partial x_{1}}},{\tfrac {\partial A_{2}}{\partial x_{1}}}-{\tfrac {\partial A_{1}}{\partial x_{2}}}\right)^{t}}$ ${\displaystyle \Phi _{B}={\rm {d}}\mathbf {A} }$ ${\displaystyle {\vec {A}}}$ ${\displaystyle \mathbf {A} :=A_{1}{\rm {d}}x_{1}+A_{2}{\rm {d}}x_{2}+A_{3}{\rm {d}}x_{3}\,.}$ ${\displaystyle (\operatorname {div} {\vec {B}}\equiv 0\,).}$

Using the Coulomb calibration or as appropriate, the following then applies to i = 1,2,3 ${\displaystyle \operatorname {div} {\vec {A}}{\stackrel {!}{=}}0}$ ${\displaystyle \operatorname {div} {\vec {j}}{\stackrel {!}{=}}0}$

${\displaystyle A_{i}({\vec {r}})=\int {\frac {\mu _{0}j_{i}({\vec {r}}^{\,'})\,\,dx_{1}'dx_{2}'dx_{3}'}{4\pi |{\vec {r}}-{\vec {r}}^{\,'}|}}\,,}$

there is a natural constant , the so-called magnetic field constant . ${\displaystyle \mu _{0}}$

In this equation u. a. remarkable that it fully corresponds to a well-known formula for the electric field , the Coulomb potential of a given charge distribution with density . At this point it is already assumed that ${\displaystyle {\vec {E}}}$ ${\displaystyle \,\phi (x_{1},x_{2},x_{3})}$ ${\displaystyle \rho (x_{1},x_{2},x_{3})}$

• ${\displaystyle {\vec {E}}}$and or ${\displaystyle {\vec {B}}}$
• ${\displaystyle \rho }$and as well ${\displaystyle {\vec {j}}}$
• ${\displaystyle \,\phi }$ and ${\displaystyle {\vec {A}}}$

can be summarized and that the relativistic invariance of Maxwell's electrodynamics results from it, see also electrodynamics .

If you give up the condition of stationarity , the time argument must be added to the space coordinates on the left side of the above equation , while on the right side the so-called "retarded time" must be added. As before, it is integrated using the three spatial coordinates . After all, the speed of light is in a vacuum . ${\displaystyle A_{i}}$ ${\displaystyle t}$ ${\displaystyle j_{i}'}$ ${\displaystyle t':=t-{\tfrac {|{\vec {r}}-{\vec {r}}^{\,'}|}{c}}}$ ${\displaystyle {\vec {r}}^{\,'}}$ ${\displaystyle c}$

Application in continuum mechanics

In continuum mechanics , the lemma is applied to tensors . B. is needed for the establishment of the compatibility conditions . The starting point is the lemma in the formulation:

${\displaystyle \operatorname {rot} {\vec {u}}={\hat {e}}_{k}\times {\frac {\partial {\vec {u}}}{\partial x_{k}}}={\vec {0}}\quad \rightarrow \quad \exists \varphi \colon {\vec {u}}=\operatorname {grad} \varphi }$		(I)

The operator "grad" forms the gradient , the vectors are the standard basis of the Cartesian coordinate system with coordinates and Einstein's summation convention was applied, according to which indices occurring twice in a product, here k, are to be summed from one to three, which is also the case in The following should be practiced. ${\displaystyle {\hat {e}}_{1,2,3}}$ ${\displaystyle x_{1,2,3}}$

Let us now be given a tensor field whose row vectors are combined with the dyadic product “ ” to form the tensor. The rotation of the transposed tensor vanishes ${\displaystyle \mathbf {T} ={\hat {e}}_{i}\otimes {\vec {t}}_{i}}$ ${\displaystyle {\vec {t}}_{1,2,3}}$ ${\displaystyle \otimes }$

${\displaystyle \operatorname {rot} (\mathbf {T} ^{\top }):={\hat {e}}_{k}\times {\frac {\partial }{\partial x_{k}}}({\vec {t}}_{i}\otimes {\hat {e}}_{i})=\left({\hat {e}}_{k}\times {\frac {\partial {\vec {t}}_{i}}{\partial x_{k}}}\right)\otimes {\hat {e}}_{i}=\mathbf {0} \quad \rightarrow \quad {\hat {e}}_{k}\times {\frac {\partial {\vec {t}}_{i}}{\partial x_{k}}}={\vec {0}}\,,\quad i=1,2,3\,,}$

so that every row vector is rotation-free. Then there is a scalar field for every row vector , the gradient of which it is: ${\displaystyle u_{i}}$

${\displaystyle {\vec {t}}_{i}=\operatorname {grad} u_{i}\quad \rightarrow \quad \mathbf {T} ={\hat {e}}_{i}\otimes {\vec {t}}_{i}={\hat {e}}_{i}\otimes \operatorname {grad} u_{i}=\operatorname {grad} {\vec {u}}\,,}$

because the gradient of the vector is formed according to: ${\displaystyle {\vec {u}}:=u_{i}{\hat {e}}_{i}}$

${\displaystyle \operatorname {grad} {\vec {u}}:={\frac {\partial u_{i}}{\partial x_{k}}}{\hat {e}}_{i}\otimes {\hat {e}}_{k}={\hat {e}}_{i}\otimes {\frac {\partial u_{i}}{\partial x_{k}}}{\hat {e}}_{k}={\hat {e}}_{i}\otimes \operatorname {grad} u_{i}\,.}$

Thus the second form of the lemma applies:

${\displaystyle \operatorname {rot} (\mathbf {T} ^{\top })=\mathbf {0} \quad \rightarrow \quad \exists {\vec {u}}\colon \mathbf {T} =\operatorname {grad} {\vec {u}}}$		(II)

If the trace of the tensor also disappears, then the vector field is free of divergence:

${\displaystyle \operatorname {Sp} (\mathbf {T} )=\operatorname {Sp} ({\hat {e}}_{i}\otimes \operatorname {grad} u_{i})={\hat {e}}_{i}\cdot {\frac {\partial u_{i}}{\partial x_{k}}}{\hat {e}}_{k}={\frac {\partial u_{i}}{\partial x_{i}}}=\operatorname {div} {\vec {u}}=0\,.}$

In this case, the Kronecker delta δ _{ij is used to} calculate :

${\displaystyle {\begin{aligned}\operatorname {rot} ({\vec {u}}\times \mathbf {I} )=&{\hat {e}}_{k}\times {\frac {\partial }{\partial x_{k}}}[u_{i}{\hat {e}}_{i}\times ({\hat {e}}_{j}\otimes {\hat {e}}_{j})]={\frac {\partial u_{i}}{\partial x_{k}}}[{\hat {e}}_{k}\times ({\hat {e}}_{i}\times {\hat {e}}_{j})]\otimes {\hat {e}}_{j}={\frac {\partial u_{i}}{\partial x_{k}}}(\delta _{jk}{\hat {e}}_{i}-\delta _{ik}{\hat {e}}_{j})\otimes {\hat {e}}_{j}\\=&{\frac {\partial u_{i}}{\partial x_{j}}}{\hat {e}}_{i}\otimes {\hat {e}}_{j}-{\frac {\partial u_{i}}{\partial x_{i}}}{\hat {e}}_{j}\otimes {\hat {e}}_{j}=\operatorname {grad} {\vec {u}}\end{aligned}}}$

and the tensor is skew symmetric : ${\displaystyle \mathbf {W} :={\vec {u}}\times \mathbf {I} }$

${\displaystyle ({\vec {u}}\times \mathbf {I} )^{\top }=(u_{i}{\hat {e}}_{i}\times {\hat {e}}_{j}\otimes {\hat {e}}_{j})^{\top }=\epsilon _{ijk}u_{i}{\hat {e}}_{j}\otimes {\hat {e}}_{k}=-u_{i}{\hat {e}}_{i}\times {\hat {e}}_{k}\otimes {\hat {e}}_{k}=-{\vec {u}}\times \mathbf {I} \,.}$

Inside is the permutation symbol . The third form of the lemma follows: ${\displaystyle \epsilon _{ijk}=({\hat {e}}_{i}\times {\hat {e}}_{j})\cdot {\hat {e}}_{k}}$

${\displaystyle \operatorname {rot} (\mathbf {T} ^{\top })=\mathbf {0} \quad {\text{und}}\quad \operatorname {Sp} (\mathbf {T} )=0\quad \rightarrow \quad \exists \mathbf {W} \colon \mathbf {T} =\operatorname {rot} \mathbf {W} \quad {\text{mit}}\quad \mathbf {W} =-\mathbf {W} ^{\top }}$		(III)

In the literature, a rotation operator is also used, which directly forms the rotation of the row vectors:

${\displaystyle \operatorname {\tilde {rot}} \mathbf {T} =\operatorname {rot} (\mathbf {T} ^{\top })\,.}$

With this operator:

${\displaystyle {\begin{array}{rrcll}{\textsf {II}}:&\operatorname {\tilde {rot}} \mathbf {T} =\mathbf {0} &\rightarrow &\exists {\vec {u}}\colon &\mathbf {T} =\operatorname {grad} {\vec {u}}\\{\textsf {III}}:&\operatorname {\tilde {rot}} \mathbf {T} =\mathbf {0} \quad {\text{und}}\quad \operatorname {Sp} (\mathbf {T} )=0&\rightarrow &\exists \mathbf {V} \colon &\mathbf {T} =\operatorname {\tilde {rot}} \mathbf {V} \quad {\text{mit}}\quad \mathbf {V} ^{\top }=-\mathbf {V} \end{array}}}$

literature

• Otto Forster : Analysis. Volume 3: Integral calculus in R ⁿ with applications. 4th edition. Vieweg + Teubner, Braunschweig et al. 2007, ISBN 978-3-528-37252-1 .
• John M. Lee: Introduction to Smooth Manifolds (= Graduate Texts in Mathematics 218). Springer-Verlag, New York NY et al. 2003, ISBN 0-387-95448-1 .
• C. Truesdell: Solid Mechanics II in S. Flügge (Ed.): Handbook of Physics , Volume VIa / 2. Springer-Verlag, 1972, ISBN 3-540-05535-5 , ISBN 0-387-05535-5 .