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An expression composed of one or more functions, their partial derivatives of various orders with respect to independent variables, and sometimes also the differentials of these variables, which is invariant with respect to certain transformations.
 
An expression composed of one or more functions, their partial derivatives of various orders with respect to independent variables, and sometimes also the differentials of these variables, which is invariant with respect to certain transformations.
  
Let a geometric object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d0322201.png" /> (cf. [[Geometric objects, theory of|Geometric objects, theory of]]) be given in a differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d0322202.png" />, the elements of which are points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d0322203.png" />. A geometric object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d0322204.png" /> of this manifold is known as a differential invariant of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d0322206.png" /> with respect to the object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d0322207.png" /> if its coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d0322208.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d0322209.png" />, are functions in the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222011.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222012.png" /> and their partial derivatives with respect to the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222014.png" />, up to the order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222015.png" />:
+
Let a geometric object $  \Omega $(
 +
cf. [[Geometric objects, theory of|Geometric objects, theory of]]) be given in a differentiable manifold $  X _ {n} $,  
 +
the elements of which are points $  ( u  ^ {1} \dots u  ^ {n} ) $.  
 +
A geometric object $  \omega $
 +
of this manifold is known as a differential invariant of order $  r $
 +
with respect to the object $  \Omega $
 +
if its coordinates $  \omega _ {A} $,  
 +
$  A = 1 \dots N $,  
 +
are functions in the coordinates $  \Omega _  \alpha  $,  
 +
$  \alpha = 1 \dots M $,  
 +
of $  \Omega $
 +
and their partial derivatives with respect to the coordinates $  u  ^ {i} $,  
 +
$  i = 1 \dots n $,  
 +
up to the order $  r $:
  
<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/d/d032/d032220/d03222016.png" /></td> </tr></table>
+
$$
 +
\omega _ {A}  = f _ {A} ( \Omega _  \alpha  , \partial  _ {i} \Omega _  \alpha  \dots \partial  _ {i _ {1}  \dots i _ {r} }  ^ {r}
 +
\Omega _  \alpha  ),
 +
$$
  
 
and have the following property of invariance with respect to coordinate transformations. In fact, under a change of coordinates
 
and have the following property of invariance with respect to coordinate transformations. In fact, under a change of coordinates
  
<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/d/d032/d032220/d03222017.png" /></td> </tr></table>
+
$$
 +
u  ^ {i}  = u  ^ {i} ( u ^ {1  ^  \prime  } \dots u ^ {n  ^  \prime  } ) ,
 +
$$
 +
 
 +
the new coordinates  $  \omega _ {A}  ^  \prime  $
 +
of  $  \omega $
 +
are expressed in terms of the new coordinates  $  \Omega _ {A}  ^  \prime  $
 +
of  $  \Omega $
 +
and their partial derivatives with respect to the new coordinates by means of the same functions  $  f _ {A} $:
 +
 
 +
$$
 +
\omega _ {A}  ^  \prime  = f _ {A} ( \Omega _ {A}  ^  \prime  , \partial  _ {i} \Omega _  \alpha  \dots \partial  _ {i _ {1}  ^  \prime  \dots i _ {n}  ^  \prime  }
 +
^ {r} \Omega _  \alpha  ) .
 +
$$
  
the new coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222019.png" /> are expressed in terms of the new coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222020.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222021.png" /> and their partial derivatives with respect to the new coordinates by means of the same functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222022.png" />:
+
For instance, let  $  \Omega $
 +
be the object of a linear affine (torsion-free) connection. The object  $  \omega $(
 +
curvature tensor)
  
<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/d/d032/d032220/d03222023.png" /></td> </tr></table>
+
$$
 +
R _ {ijk} ^ { l }  = \partial  _ {i} \Gamma _ {jk}  ^ {l}
 +
- \partial  _ {j} \Gamma _ {ik}  ^ {l} + \Gamma _ {is}  ^ {l} \Gamma _ {jk}  ^ {s} - \Gamma _ {js}  ^ {l} \Gamma _ {ik}  ^ {s}
 +
$$
  
For instance, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222024.png" /> be the object of a linear affine (torsion-free) connection. The object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222025.png" /> (curvature tensor)
+
is a tensor differential invariant of the first order with respect to the Christoffel symbols  $  \Gamma _ {ij}  ^ {k} $(
 +
cf. [[Christoffel symbol|Christoffel symbol]]).
  
<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/d/d032/d032220/d03222026.png" /></td> </tr></table>
+
Let there be given in  $  X _ {n} $
 +
a group (pseudo-group)  $  G $
 +
of point transformations
  
is a tensor differential invariant of the first order with respect to the Christoffel symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222027.png" /> (cf. [[Christoffel symbol|Christoffel symbol]]).
+
$$ \tag{1 }
 +
u  ^ {i}  = f ^ { i } ( \overline{\mathbf u}\; {}  ^ {1} \dots \overline{\mathbf u}\; {}  ^ {n} )
 +
$$
  
Let there be given in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222028.png" /> a group (pseudo-group) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222029.png" /> of point transformations
+
and let  $  M _ {h} $
 +
be a submanifold of  $  X _ {n} $
 +
of dimension  $  h $:
  
<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/d/d032/d032220/d03222030.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{2 }
 +
u  ^ {i}  = \phi  ^ {i} ( t  ^ {1} \dots t  ^ {n} ) ,
 +
$$
  
and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222031.png" /> be a submanifold of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222032.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222033.png" />:
+
the parameters of which are subject to transformations of the infinite group  $  G _  \infty  $:
  
<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/d/d032/d032220/d03222034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$
 +
t  ^  \alpha  = \psi  ^  \alpha  ( \overline{t}\; {}  ^ {1*} \dots \overline{t}\; {}  ^ {n*} ) .
 +
$$
  
the parameters of which are subject to transformations of the infinite group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222035.png" />:
+
A geometric differential invariant of order  $  r $
 +
of the manifold  $  M _ {h} $
 +
with respect to the group (pseudo-group)  $  G $
 +
is the name of a function of the coordinates  $  u  ^ {i} $
 +
of a point of $  M _ {h} $
 +
and their partial derivatives up to order  $  r $
 +
with respect to the parameters  $  t  ^  \alpha  $:
  
<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/d/d032/d032220/d03222036.png" /></td> </tr></table>
+
$$ \tag{3 }
 +
F \left ( u  ^ {i} ,
 +
\frac{\partial  u  ^ {i} }{\partial  t  ^  \alpha  }
  
A geometric differential invariant of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222037.png" /> of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222038.png" /> with respect to the group (pseudo-group) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222039.png" /> is the name of a function of the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222040.png" /> of a point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222041.png" /> and their partial derivatives up to order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222042.png" /> with respect to the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222043.png" />:
+
\dots
 +
\frac{\partial  ^ {r} u  ^ {i} }{\partial  t ^ {\alpha _ {1} }
 +
{} \dots \partial  t ^ {\alpha _ {r} } }
 +
\right ) ,
 +
$$
  
<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/d/d032/d032220/d03222044.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
which is invariant with respect to the transformations (1) and (2). In fact, if  $  u  ^ {i} $
 +
are substituted in (3) according to the formulas (1), while the partial derivatives of  $  u  ^ {i} $
 +
with respect to  $  t  ^  \alpha  $
 +
are replaced by their expressions in terms of derivatives of  $  \overline{\mathbf u}\; {}  ^ {i} $
 +
with respect to  $  \overline{\mathbf t}\; {} ^ {\alpha  ^ {*} } $,
 +
one obtains the same function  $  F $
 +
in  $  \overline{\mathbf u}\; {}  ^ {i} $
 +
and their derivatives with respect to  $  \overline{\mathbf t}\; {} ^ {\alpha  ^ {*} } $:
  
which is invariant with respect to the transformations (1) and (2). In fact, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222045.png" /> are substituted in (3) according to the formulas (1), while the partial derivatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222046.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222047.png" /> are replaced by their expressions in terms of derivatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222048.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222049.png" />, one obtains the same function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222050.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222051.png" /> and their derivatives with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222052.png" />:
+
$$
 +
F \left ( u  ^ {i} ,  
 +
\frac{\partial  {\overline{\mathbf u}\; } {}  ^ {i} }{\partial  {\overline{\mathbf t}\; } {} ^ {\alpha _ {1} } }
  
<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/d/d032/d032220/d03222053.png" /></td> </tr></table>
+
\dots
 +
\frac{\partial  ^ {r} {\overline{\mathbf u}\; } {}  ^ {i} }{\partial  {\overline{\mathbf t}\; } {} ^
 +
{\alpha _ {r} } \dots \partial  {\overline{\mathbf t}\; } {} ^ {\alpha _ {r} } }
 +
\right ) .
 +
$$
  
If the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222054.png" /> are homogeneous, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222055.png" /> should also be invariant with respect to the transformations
+
If the coordinates $  u  ^ {i} $
 +
are homogeneous, then $  F $
 +
should also be invariant with respect to the transformations
  
<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/d/d032/d032220/d03222056.png" /></td> </tr></table>
+
$$
 +
u ^ {*  _ {i} }  = \lambda ( t  ^ {1} \dots t  ^ {n} ) u  ^ {i} ,\ \
 +
\lambda \neq 0 .
 +
$$
  
In the definition of a geometric differential invariant, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032220/d03222057.png" /> may be replaced by a geometric object. If this object is a covariant (contravariant) vector, it is named covariant (contravariant).
+
In the definition of a geometric differential invariant, $  F $
 +
may be replaced by a geometric object. If this object is a covariant (contravariant) vector, it is named covariant (contravariant).
  
 
If the vanishing of some object is invariant, the object is named a relative differential invariant.
 
If the vanishing of some object is invariant, the object is named a relative differential invariant.

Latest revision as of 19:35, 5 June 2020


An expression composed of one or more functions, their partial derivatives of various orders with respect to independent variables, and sometimes also the differentials of these variables, which is invariant with respect to certain transformations.

Let a geometric object $ \Omega $( cf. Geometric objects, theory of) be given in a differentiable manifold $ X _ {n} $, the elements of which are points $ ( u ^ {1} \dots u ^ {n} ) $. A geometric object $ \omega $ of this manifold is known as a differential invariant of order $ r $ with respect to the object $ \Omega $ if its coordinates $ \omega _ {A} $, $ A = 1 \dots N $, are functions in the coordinates $ \Omega _ \alpha $, $ \alpha = 1 \dots M $, of $ \Omega $ and their partial derivatives with respect to the coordinates $ u ^ {i} $, $ i = 1 \dots n $, up to the order $ r $:

$$ \omega _ {A} = f _ {A} ( \Omega _ \alpha , \partial _ {i} \Omega _ \alpha \dots \partial _ {i _ {1} \dots i _ {r} } ^ {r} \Omega _ \alpha ), $$

and have the following property of invariance with respect to coordinate transformations. In fact, under a change of coordinates

$$ u ^ {i} = u ^ {i} ( u ^ {1 ^ \prime } \dots u ^ {n ^ \prime } ) , $$

the new coordinates $ \omega _ {A} ^ \prime $ of $ \omega $ are expressed in terms of the new coordinates $ \Omega _ {A} ^ \prime $ of $ \Omega $ and their partial derivatives with respect to the new coordinates by means of the same functions $ f _ {A} $:

$$ \omega _ {A} ^ \prime = f _ {A} ( \Omega _ {A} ^ \prime , \partial _ {i} \Omega _ \alpha \dots \partial _ {i _ {1} ^ \prime \dots i _ {n} ^ \prime } ^ {r} \Omega _ \alpha ) . $$

For instance, let $ \Omega $ be the object of a linear affine (torsion-free) connection. The object $ \omega $( curvature tensor)

$$ R _ {ijk} ^ { l } = \partial _ {i} \Gamma _ {jk} ^ {l} - \partial _ {j} \Gamma _ {ik} ^ {l} + \Gamma _ {is} ^ {l} \Gamma _ {jk} ^ {s} - \Gamma _ {js} ^ {l} \Gamma _ {ik} ^ {s} $$

is a tensor differential invariant of the first order with respect to the Christoffel symbols $ \Gamma _ {ij} ^ {k} $( cf. Christoffel symbol).

Let there be given in $ X _ {n} $ a group (pseudo-group) $ G $ of point transformations

$$ \tag{1 } u ^ {i} = f ^ { i } ( \overline{\mathbf u}\; {} ^ {1} \dots \overline{\mathbf u}\; {} ^ {n} ) $$

and let $ M _ {h} $ be a submanifold of $ X _ {n} $ of dimension $ h $:

$$ \tag{2 } u ^ {i} = \phi ^ {i} ( t ^ {1} \dots t ^ {n} ) , $$

the parameters of which are subject to transformations of the infinite group $ G _ \infty $:

$$ t ^ \alpha = \psi ^ \alpha ( \overline{t}\; {} ^ {1*} \dots \overline{t}\; {} ^ {n*} ) . $$

A geometric differential invariant of order $ r $ of the manifold $ M _ {h} $ with respect to the group (pseudo-group) $ G $ is the name of a function of the coordinates $ u ^ {i} $ of a point of $ M _ {h} $ and their partial derivatives up to order $ r $ with respect to the parameters $ t ^ \alpha $:

$$ \tag{3 } F \left ( u ^ {i} , \frac{\partial u ^ {i} }{\partial t ^ \alpha } \dots \frac{\partial ^ {r} u ^ {i} }{\partial t ^ {\alpha _ {1} } {} \dots \partial t ^ {\alpha _ {r} } } \right ) , $$

which is invariant with respect to the transformations (1) and (2). In fact, if $ u ^ {i} $ are substituted in (3) according to the formulas (1), while the partial derivatives of $ u ^ {i} $ with respect to $ t ^ \alpha $ are replaced by their expressions in terms of derivatives of $ \overline{\mathbf u}\; {} ^ {i} $ with respect to $ \overline{\mathbf t}\; {} ^ {\alpha ^ {*} } $, one obtains the same function $ F $ in $ \overline{\mathbf u}\; {} ^ {i} $ and their derivatives with respect to $ \overline{\mathbf t}\; {} ^ {\alpha ^ {*} } $:

$$ F \left ( u ^ {i} , \frac{\partial {\overline{\mathbf u}\; } {} ^ {i} }{\partial {\overline{\mathbf t}\; } {} ^ {\alpha _ {1} } } \dots \frac{\partial ^ {r} {\overline{\mathbf u}\; } {} ^ {i} }{\partial {\overline{\mathbf t}\; } {} ^ {\alpha _ {r} } \dots \partial {\overline{\mathbf t}\; } {} ^ {\alpha _ {r} } } \right ) . $$

If the coordinates $ u ^ {i} $ are homogeneous, then $ F $ should also be invariant with respect to the transformations

$$ u ^ {* _ {i} } = \lambda ( t ^ {1} \dots t ^ {n} ) u ^ {i} ,\ \ \lambda \neq 0 . $$

In the definition of a geometric differential invariant, $ F $ may be replaced by a geometric object. If this object is a covariant (contravariant) vector, it is named covariant (contravariant).

If the vanishing of some object is invariant, the object is named a relative differential invariant.

References

[1] T.Y. Thomas, "The differential invariants of generalized spaces" , Cambridge Univ. Press (1934)
[2] R. Weitzenböck, "Invariantentheorie" , Noordhoff (1923)
How to Cite This Entry:
Differential invariant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Differential_invariant&oldid=46692
This article was adapted from an original article by V.I. Shulikovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article