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Difference between revisions of "Darboux theorem"

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{{DEF}}
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'''Darboux theorem may''' may refer to one of the following assertions:
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* Darboux theorem on local canonical coordinates for symplectic structure;
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* Darboux theorem on intermediate values of the derivative of a function of one variable.
  
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== Darboux theorems for symplectic structure ===
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{{MSC|37Jxx,53Dxx}}
  
If a real-valued function has a finite derivative at each point of an interval on the real axis, then, if the derivative assumes any two values on this interval, this derivative also assumes all the intermediate values on it.
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Recall that a [[symplectic structure]] on an even-dimensional manifold $M^{2n}$ is a closed nondegenerate 2-form $\omega$:
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$$
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\omega\in\varLambda^2(M),\qquad \rd \omega=0,\qquad \forall v\in T_p M\quad \exists w\in T_p M:\ \omega_p(v,w)\ne0.
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$$
  
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The matrix $S(z)$ of a symplectic structure, $S_{ij}(z)=\omega(\frac{\partial}{\partial z_i},\frac{\partial}{\partial z_i})$  in any local coordinate system $(z_1,\dots,z_{2n})$ is antisymmetric and nondegenerate: $\omega=\frac12\sum S_{ij}(z)\,\rd z_i\land \rd z_j$.
  
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The ''standard symplectic structure'' on $\R^{2n}$ in the ''standard canonical coordinates'' $(x_1,\dots,x_n,p_1,\dots,p_n)$ is given by the form
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$$
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\omega=\sum_{i=1}^n \rd x_i\land \rd p_i.
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$$
  
====Comments====
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== Darboux therem for intermediate values of differentiable functions ==
This is a kind of intermediate value theorem (for the derivative rather than the function). The following theorem holds for functions themselves: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030190/d0301901.png" /> is a [[Continuous mapping|continuous mapping]] between metric spaces, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030190/d0301902.png" /> is a [[Connected set|connected set]], then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030190/d0301903.png" /> is connected. (See, e.g., [[#References|[a1]]].)
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,  "Principles of mathematical analysis" , McGraw-Hill  (1953)</TD></TR></table>
 

Revision as of 12:48, 29 April 2012

Darboux theorem may may refer to one of the following assertions:

  • Darboux theorem on local canonical coordinates for symplectic structure;
  • Darboux theorem on intermediate values of the derivative of a function of one variable.

Darboux theorems for symplectic structure =

2020 Mathematics Subject Classification: Primary: 37Jxx,53Dxx [MSN][ZBL]

Recall that a symplectic structure on an even-dimensional manifold $M^{2n}$ is a closed nondegenerate 2-form $\omega$: $$ \omega\in\varLambda^2(M),\qquad \rd \omega=0,\qquad \forall v\in T_p M\quad \exists w\in T_p M:\ \omega_p(v,w)\ne0. $$

The matrix $S(z)$ of a symplectic structure, $S_{ij}(z)=\omega(\frac{\partial}{\partial z_i},\frac{\partial}{\partial z_i})$ in any local coordinate system $(z_1,\dots,z_{2n})$ is antisymmetric and nondegenerate: $\omega=\frac12\sum S_{ij}(z)\,\rd z_i\land \rd z_j$.

The standard symplectic structure on $\R^{2n}$ in the standard canonical coordinates $(x_1,\dots,x_n,p_1,\dots,p_n)$ is given by the form $$ \omega=\sum_{i=1}^n \rd x_i\land \rd p_i. $$

Darboux therem for intermediate values of differentiable functions

How to Cite This Entry:
Darboux theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Darboux_theorem&oldid=25693
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article