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''closed system (of differential equations)''
 
''closed system (of differential equations)''
  
 
A system of first-order partial differential equations:
 
A system of first-order partial differential equations:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c0238901.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
F _ {i} ( x, u , p)  = 0 ,\ \
 +
1 \leq  i \leq  m,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c0238902.png" /></td> </tr></table>
+
$$
 +
= ( x _ {1} \dots x _ {n} ),\  u  = u( x _ {1} \dots x _ {n} ),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c0238903.png" /></td> </tr></table>
+
$$
 +
= ( p _ {1} \dots p _ {n} )  = \left (
 +
\frac{\partial  u }{\partial  x _ {1} }
 +
\dots
 +
\frac{\partial  u }{\partial  x _ {n} }
 +
\right ) ,
 +
$$
  
with the following property: For any set of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c0238904.png" /> satisfying (1), the equation
+
with the following property: For any set of numbers $  ( x, u , p) $
 +
satisfying (1), the equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c0238905.png" /></td> </tr></table>
+
$$
 +
F _ {ij} ( x, u , p)  = 0,\ \
 +
1 \leq  i, j \leq  m,
 +
$$
  
is valid, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c0238906.png" /> are the [[Jacobi brackets|Jacobi brackets]].
+
is valid, where $  F _ {ij} = [ F _ {i} , F _ {j} ] $
 +
are the [[Jacobi brackets|Jacobi brackets]].
  
The completeness condition may be formulated somewhat differently for a linear homogeneous system. The Jacobi brackets in that case are linear in the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c0238907.png" />; if the system is written in the form
+
The completeness condition may be formulated somewhat differently for a linear homogeneous system. The Jacobi brackets in that case are linear in the variables $  p = ( p _ {1} \dots p _ {n} ) $;  
 +
if the system is written in the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c0238908.png" /></td> </tr></table>
+
$$
 +
P _ {i} ( u)  = 0,\ \
 +
1 \leq  i \leq  m,
 +
$$
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c0238909.png" /> are linear first-order differential operators, then these brackets correspond to the commutators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389010.png" />. The system is complete if all commutators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389011.png" /> can be expressed as linear combinations of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389012.png" />'s with coefficients depending only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389013.png" />.
+
where the $  P _ {i} $
 +
are linear first-order differential operators, then these brackets correspond to the commutators $  [ P _ {i} , P _ {j} ] = P _ {i} P _ {j} - P _ {j} P _ {i} $.  
 +
The system is complete if all commutators $  [ P _ {i} , P _ {j} ] $
 +
can be expressed as linear combinations of the $  P _ {k} $'
 +
s with coefficients depending only on $  x = ( x _ {1} \dots x _ {n} ) $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389014.png" /> is the joint solution of the two equations
+
If $  u = u( x) $
 +
is the joint solution of the two equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389015.png" /></td> </tr></table>
+
$$
 +
F _ {i} ( x, u , p)  = 0,\ \
 +
F _ {j} ( x, u , p)  = 0,
 +
$$
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389016.png" /> is also a solution of the equation
+
then $  u $
 +
is also a solution of the equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
[ F _ {i} , F _ {j} ] ( x, u , p)  = 0.
 +
$$
  
 
An arbitrary system of the form (1) is usually extended to a complete one by adding new independent equations to it that have been obtained from the old ones by means of the formation of Jacobi brackets. In this extension, in accordance with (2), none of the solutions will be lost if the system is generally solvable.
 
An arbitrary system of the form (1) is usually extended to a complete one by adding new independent equations to it that have been obtained from the old ones by means of the formation of Jacobi brackets. In this extension, in accordance with (2), none of the solutions will be lost if the system is generally solvable.
  
A property of the system is that it is completely invariant under those non-singular transformations of the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389018.png" /> for which the meaning of the differential equations is retained. These transformations include, for example, diffeomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389020.png" />, and also transformations of the following type. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389021.png" /> be a smooth mapping such that
+
A property of the system is that it is completely invariant under those non-singular transformations of the variables $  ( x, u , F) $
 +
for which the meaning of the differential equations is retained. These transformations include, for example, diffeomorphisms $  g( y) $,  
 +
$  y = ( y _ {1} \dots y _ {n} ) $,  
 +
and also transformations of the following type. Let $  H: \mathbf R  ^ {2n+} 1+ m \rightarrow \mathbf R  ^ {m} $
 +
be a smooth mapping such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389022.png" /></td> </tr></table>
+
$$
 +
= x,\  q  = p,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389023.png" /></td> </tr></table>
+
$$
 +
= u ,\  t  = H( x, u , p, s),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389024.png" /></td> </tr></table>
+
$$
 +
= ( s _ {1} \dots s _ {m} ),\  t  = ( t _ {1} \dots t _ {m} ),
 +
$$
  
is a diffeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389025.png" />. This transforms the system (1) into a system
+
is a diffeomorphism $  \mathbf R  ^ {2n+} 1+ m \rightarrow \mathbf R  ^ {2n+} 1+ m $.  
 +
This transforms the system (1) into a system
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389026.png" /></td> </tr></table>
+
$$
 +
G _ {i} ( x, u , p)  = \
 +
H _ {i} ( x, u , p, F )  = 0,\ \
 +
1 \leq  i \leq  m.
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''2. Partielle Differentialgleichungen erster Ordnung für die gesuchte Funktion''' , Akad. Verlagsgesell.  (1944)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.M. Gyunter,  "Integrating first-order partial differential equations" , Leningrad-Moscow  (1934)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Carathéodory,  "Calculus of variations and partial differential equations of the first order" , '''1''' , Holden-Day  (1965)  (Translated from German)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E. Goursat,  "Leçons sur l'intégration des équations aux dérivées partielles du premier ordre" , Hermann  (1891)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''2. Partielle Differentialgleichungen erster Ordnung für die gesuchte Funktion''' , Akad. Verlagsgesell.  (1944)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.M. Gyunter,  "Integrating first-order partial differential equations" , Leningrad-Moscow  (1934)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  C. Carathéodory,  "Calculus of variations and partial differential equations of the first order" , '''1''' , Holden-Day  (1965)  (Translated from German)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E. Goursat,  "Leçons sur l'intégration des équations aux dérivées partielles du premier ordre" , Hermann  (1891)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
A system (1) can be rewritten in the form:
 
A system (1) can be rewritten in the form:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389027.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
$$ \tag{a1 }
 +
\widetilde{F}  _ {i} ( \widetilde{x}  , \widetilde{p}  )  = 0,\ \
 +
i = 1 \dots m,
 +
$$
 +
 
 +
$$
 +
\widetilde{x}  \in  \mathbf R ^ {n + 1 } , \widetilde{p}  = \
 +
\left (  
 +
\frac{\partial  \widetilde{u}  }{\partial  x _ {1} }
 +
\dots
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389028.png" /></td> </tr></table>
+
\frac{\partial  \widetilde{u}  }{\partial  x _ {n + 1 }  }
 +
\right )  \in  \mathbf R ^ {n + 1 } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389029.png" /> defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389030.png" /> implicitly:
+
where $  \widetilde{u}  $
 +
defines $  u $
 +
implicitly:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389031.png" /></td> </tr></table>
+
$$
 +
\widetilde{u}
 +
( x _ {1} \dots x _ {n} , u )  = 0,
 +
$$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389032.png" />. (The system (1) may admit of singular solutions not represented by (a1), see [[#References|[1]]].) The Jacobi brackets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389033.png" /> thus reduce to the [[Poisson brackets|Poisson brackets]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389034.png" />.
+
and $  u = x _ {n + 1 }  $.  
 +
(The system (1) may admit of singular solutions not represented by (a1), see [[#References|[1]]].) The Jacobi brackets $  [ \widetilde{F}  _ {i} , \widetilde{F}  _ {j} ] $
 +
thus reduce to the [[Poisson brackets|Poisson brackets]] $  \{ \widetilde{F}  _ {i} , \widetilde{F}  _ {j} \} $.
  
The system (a1) defines level sets of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389036.png" />, on the cotangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389037.png" />. (a1) is complete if the Poisson brackets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389038.png" /> vanish on the intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389039.png" />. (a1) is in involution in a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389040.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389041.png" /> if the Poisson brackets vanish identically on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389042.png" /> and, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389043.png" /> are linearly independent on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389044.png" />.
+
The system (a1) defines level sets of the functions $  \widetilde{F}  _ {i} $,  
 +
$  i = 1 \dots m $,  
 +
on the cotangent bundle $  T  ^ {*} ( \mathbf R ^ {n + 1 } ) $.  
 +
(a1) is complete if the Poisson brackets $  \{ \widetilde{F}  _ {i} , \widetilde{F}  _ {j} \} _ {i, j = 1 }  ^ {m} $
 +
vanish on the intersection $  M = \cap _ {i} \{ {( \widetilde{x}  , \widetilde{p}  ) } : {\widetilde{F}  _ {i} ( \widetilde{x}  , \widetilde{p}  ) = 0 } \} \subset  T  ^ {*} ( \mathbf R ^ {n + 1 } ) $.  
 +
(a1) is in involution in a neighbourhood $  U $
 +
of $  ( \widetilde{x}  , \widetilde{p}  ) \in T  ^ {*} ( \mathbf R ^ {n + 1 } ) $
 +
if the Poisson brackets vanish identically on $  U $
 +
and, moreover, $  \{ d \widetilde{F}  _ {i} \} _ {1}  ^ {m} $
 +
are linearly independent on $  U $.
  
(a1) is in involution in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389045.png" /> if it is complete and the linear independency condition holds on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023890/c02389046.png" />. Being in involution is a necessary condition for the existence of a solution of the system (1) (or (a1)), cf. [[Darboux theorem|Darboux theorem]] and [[#References|[a1]]], Sect. 21.1. As suggested in the main article, these considerations generalize to systems defined on general differentiable manifolds.
+
(a1) is in involution in a neighbourhood of $  M $
 +
if it is complete and the linear independency condition holds on $  M $.  
 +
Being in involution is a necessary condition for the existence of a solution of the system (1) (or (a1)), cf. [[Darboux theorem|Darboux theorem]] and [[#References|[a1]]], Sect. 21.1. As suggested in the main article, these considerations generalize to systems defined on general differentiable manifolds.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.V. Hörmander,  "The analysis of linear partial differential operators" , '''3''' , Springer  (1985)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.V. Hörmander,  "The analysis of linear partial differential operators" , '''3''' , Springer  (1985)</TD></TR></table>

Latest revision as of 17:45, 4 June 2020


closed system (of differential equations)

A system of first-order partial differential equations:

$$ \tag{1 } F _ {i} ( x, u , p) = 0 ,\ \ 1 \leq i \leq m, $$

$$ x = ( x _ {1} \dots x _ {n} ),\ u = u( x _ {1} \dots x _ {n} ), $$

$$ p = ( p _ {1} \dots p _ {n} ) = \left ( \frac{\partial u }{\partial x _ {1} } \dots \frac{\partial u }{\partial x _ {n} } \right ) , $$

with the following property: For any set of numbers $ ( x, u , p) $ satisfying (1), the equation

$$ F _ {ij} ( x, u , p) = 0,\ \ 1 \leq i, j \leq m, $$

is valid, where $ F _ {ij} = [ F _ {i} , F _ {j} ] $ are the Jacobi brackets.

The completeness condition may be formulated somewhat differently for a linear homogeneous system. The Jacobi brackets in that case are linear in the variables $ p = ( p _ {1} \dots p _ {n} ) $; if the system is written in the form

$$ P _ {i} ( u) = 0,\ \ 1 \leq i \leq m, $$

where the $ P _ {i} $ are linear first-order differential operators, then these brackets correspond to the commutators $ [ P _ {i} , P _ {j} ] = P _ {i} P _ {j} - P _ {j} P _ {i} $. The system is complete if all commutators $ [ P _ {i} , P _ {j} ] $ can be expressed as linear combinations of the $ P _ {k} $' s with coefficients depending only on $ x = ( x _ {1} \dots x _ {n} ) $.

If $ u = u( x) $ is the joint solution of the two equations

$$ F _ {i} ( x, u , p) = 0,\ \ F _ {j} ( x, u , p) = 0, $$

then $ u $ is also a solution of the equation

$$ \tag{2 } [ F _ {i} , F _ {j} ] ( x, u , p) = 0. $$

An arbitrary system of the form (1) is usually extended to a complete one by adding new independent equations to it that have been obtained from the old ones by means of the formation of Jacobi brackets. In this extension, in accordance with (2), none of the solutions will be lost if the system is generally solvable.

A property of the system is that it is completely invariant under those non-singular transformations of the variables $ ( x, u , F) $ for which the meaning of the differential equations is retained. These transformations include, for example, diffeomorphisms $ g( y) $, $ y = ( y _ {1} \dots y _ {n} ) $, and also transformations of the following type. Let $ H: \mathbf R ^ {2n+} 1+ m \rightarrow \mathbf R ^ {m} $ be a smooth mapping such that

$$ y = x,\ q = p, $$

$$ v = u ,\ t = H( x, u , p, s), $$

$$ s = ( s _ {1} \dots s _ {m} ),\ t = ( t _ {1} \dots t _ {m} ), $$

is a diffeomorphism $ \mathbf R ^ {2n+} 1+ m \rightarrow \mathbf R ^ {2n+} 1+ m $. This transforms the system (1) into a system

$$ G _ {i} ( x, u , p) = \ H _ {i} ( x, u , p, F ) = 0,\ \ 1 \leq i \leq m. $$

References

[1] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 2. Partielle Differentialgleichungen erster Ordnung für die gesuchte Funktion , Akad. Verlagsgesell. (1944)
[2] N.M. Gyunter, "Integrating first-order partial differential equations" , Leningrad-Moscow (1934) (In Russian)
[3] C. Carathéodory, "Calculus of variations and partial differential equations of the first order" , 1 , Holden-Day (1965) (Translated from German)
[4] E. Goursat, "Leçons sur l'intégration des équations aux dérivées partielles du premier ordre" , Hermann (1891)

Comments

A system (1) can be rewritten in the form:

$$ \tag{a1 } \widetilde{F} _ {i} ( \widetilde{x} , \widetilde{p} ) = 0,\ \ i = 1 \dots m, $$

$$ \widetilde{x} \in \mathbf R ^ {n + 1 } , \widetilde{p} = \ \left ( \frac{\partial \widetilde{u} }{\partial x _ {1} } \dots \frac{\partial \widetilde{u} }{\partial x _ {n + 1 } } \right ) \in \mathbf R ^ {n + 1 } , $$

where $ \widetilde{u} $ defines $ u $ implicitly:

$$ \widetilde{u} ( x _ {1} \dots x _ {n} , u ) = 0, $$

and $ u = x _ {n + 1 } $. (The system (1) may admit of singular solutions not represented by (a1), see [1].) The Jacobi brackets $ [ \widetilde{F} _ {i} , \widetilde{F} _ {j} ] $ thus reduce to the Poisson brackets $ \{ \widetilde{F} _ {i} , \widetilde{F} _ {j} \} $.

The system (a1) defines level sets of the functions $ \widetilde{F} _ {i} $, $ i = 1 \dots m $, on the cotangent bundle $ T ^ {*} ( \mathbf R ^ {n + 1 } ) $. (a1) is complete if the Poisson brackets $ \{ \widetilde{F} _ {i} , \widetilde{F} _ {j} \} _ {i, j = 1 } ^ {m} $ vanish on the intersection $ M = \cap _ {i} \{ {( \widetilde{x} , \widetilde{p} ) } : {\widetilde{F} _ {i} ( \widetilde{x} , \widetilde{p} ) = 0 } \} \subset T ^ {*} ( \mathbf R ^ {n + 1 } ) $. (a1) is in involution in a neighbourhood $ U $ of $ ( \widetilde{x} , \widetilde{p} ) \in T ^ {*} ( \mathbf R ^ {n + 1 } ) $ if the Poisson brackets vanish identically on $ U $ and, moreover, $ \{ d \widetilde{F} _ {i} \} _ {1} ^ {m} $ are linearly independent on $ U $.

(a1) is in involution in a neighbourhood of $ M $ if it is complete and the linear independency condition holds on $ M $. Being in involution is a necessary condition for the existence of a solution of the system (1) (or (a1)), cf. Darboux theorem and [a1], Sect. 21.1. As suggested in the main article, these considerations generalize to systems defined on general differentiable manifolds.

References

[a1] L.V. Hörmander, "The analysis of linear partial differential operators" , 3 , Springer (1985)
How to Cite This Entry:
Complete system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_system&oldid=46423
This article was adapted from an original article by A.P. Soldatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article