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''nabla operator, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h0462402.png" />-operator, Hamiltonian''
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A symbolic first-order differential operator, used for the notation of one of the principal differential operations of vector analysis. In a rectangular Cartesian coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h0462403.png" /> with unit vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h0462404.png" />, the Hamilton operator has the form
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{{TEX|auto}}
 +
{{TEX|done}}
  
<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/h/h046/h046240/h0462405.png" /></td> </tr></table>
+
''nabla operator,  $  \nabla $-
 +
operator, Hamiltonian''
  
The application of the Hamilton operator to a scalar function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h0462406.png" />, which is understood as multiplication of the  "vector" <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h0462407.png" /> by the scalar <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h0462408.png" />, yields the [[Gradient|gradient]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h0462409.png" />:
+
A symbolic first-order differential operator, used for the notation of one of the principal differential operations of vector analysis. In a rectangular Cartesian coordinate system $ x = ( x _ {1} \dots x _ {n} ) $
 +
with unit vectors  $  \mathbf e _ {1} \dots \mathbf e _ {n} $,
 +
the Hamilton operator has 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/h/h046/h046240/h04624010.png" /></td> </tr></table>
+
$$
 +
\nabla  = \
 +
\sum _ {j = 1 } ^ { n }
 +
\mathbf e _ {j}
 +
\frac \partial {\partial  x _ {j} }
 +
.
 +
$$
  
i.e. the vector with components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h04624011.png" />.
+
The application of the Hamilton operator to a scalar function  $  f $,
 +
which is understood as multiplication of the  "vector"   $  \nabla $
 +
by the scalar  $  f ( x) $,
 +
yields the [[Gradient|gradient]] of  $  f $:
  
The scalar product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h04624012.png" /> with a field vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h04624013.png" /> yields the [[Divergence|divergence]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h04624014.png" />:
+
$$
 +
\mathop{\rm grad}  f  = \
 +
\nabla f  = \
 +
\sum _ {j = 1 } ^ { n }
 +
\mathbf e _ {j}
 +
\frac{\partial  f }{\partial  x _ {j} }
 +
,
 +
$$
  
<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/h/h046/h046240/h04624015.png" /></td> </tr></table>
+
i.e. the vector with components  $  ( \partial  f / \partial  x _ {1} \dots \partial  f / \partial  x _ {n} ) $.
  
The vector product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h04624016.png" /> with the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h04624017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h04624018.png" />, yields the [[Curl|curl]] (rotation, abbreviated by rot) of the fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h04624019.png" />, i.e. the vector
+
The scalar product of $  \nabla $
 +
with a field vector  $  \mathbf a = ( a _ {1} \dots a _ {n} ) $
 +
yields the [[Divergence|divergence]] of $  \mathbf a $:
  
<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/h/h046/h046240/h04624020.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm div}  \mathbf a  = \
 +
\nabla \mathbf a  = \
 +
\sum _ {j = 1 } ^ { n }
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046240/h04624021.png" />,
+
\frac{\partial  a _ {j} }{\partial  x _ {j} }
 +
.
 +
$$
  
<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/h/h046/h046240/h04624022.png" /></td> </tr></table>
+
The vector product of  $  \nabla $
 +
with the vectors  $  \mathbf a _ {j} = ( a _ {j1} \dots a _ {jn} ) $,
 +
$  j = 1 \dots n - 2 $,
 +
yields the [[Curl|curl]] (rotation, abbreviated by rot) of the fields  $  \mathbf a _ {1} \dots \mathbf a _ {n-} 2 $,
 +
i.e. the vector
  
<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/h/h046/h046240/h04624023.png" /></td> </tr></table>
+
$$
 +
[ \nabla , \mathbf a _ {1} \dots \mathbf a _ {n - 2 }  ]  = \
 +
\left |
 +
 
 +
\begin{array}{cccc}
 +
\mathbf e _ {1}  &\mathbf e _ {2}  &\dots  &\mathbf e _ {n}  \\
 +
{
 +
\frac \partial {\partial  x _ {1} }
 +
}  &{
 +
\frac \partial {\partial  x _ {2} }
 +
}  &\dots  &{
 +
\frac \partial {\partial  x _ {n} }
 +
}  \\
 +
a _ {11}  &a _ {12}  &\dots  &a _ {1n}  \\
 +
\cdot  &\cdot  &{}  &\cdot  \\
 +
\cdot  &\cdot  &{}  &\cdot  \\
 +
a _ {n - 2,1 }  &a _ {n - 2,2 }  &\dots  &a _ {n - 2,n }  \\
 +
\end{array}
 +
 
 +
\right | .
 +
$$
 +
 
 +
If  $  n = 3 $,
 +
 
 +
$$
 +
[ \nabla , \mathbf a ]  = \nabla \times \mathbf a  = \
 +
\mathop{\rm rot}  \mathbf a  = \
 +
\left (
 +
 
 +
\frac{\partial  a _ {3} }{\partial  x _ {2} }
 +
-
 +
 
 +
\frac{\partial  a _ {2} }{\partial  x _ {3} }
 +
 
 +
\right )
 +
\mathbf e _ {1} +
 +
$$
 +
 
 +
$$
 +
+
 +
\left (
 +
\frac{\partial  a _ {1} }{\partial  x _ {3}  }
 +
-
 +
\frac{\partial  a _ {3} }{\partial  x _ {1} }
 +
\right
 +
) \mathbf e _ {2} + \left (
 +
\frac{\partial  a _ {2} }{\partial  x _ {1} }
 +
-
 +
\frac{\partial  a _ {1} }{\partial  x _ {2} }
 +
\right ) \mathbf e _ {3} .
 +
$$
  
 
The scalar square of the Hamilton operator yields the [[Laplace operator|Laplace operator]]:
 
The scalar square of the Hamilton operator yields the [[Laplace operator|Laplace operator]]:
  
<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/h/h046/h046240/h04624024.png" /></td> </tr></table>
+
$$
 +
\Delta  = \
 +
\nabla \cdot \nabla  = \
 +
\sum _ {j = 1 } ^ { n }
 +
 
 +
\frac{\partial  ^ {2} }{\partial  x _ {j}  ^ {2} }
 +
.
 +
$$
  
 
The following relations are valid:
 
The following relations are valid:
  
<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/h/h046/h046240/h04624025.png" /></td> </tr></table>
+
$$
 +
[ \nabla , \nabla \phi ]  = \
 +
\mathop{\rm rot}  \mathop{\rm grad}  \phi  = 0,
 +
$$
  
<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/h/h046/h046240/h04624026.png" /></td> </tr></table>
+
$$
 +
\nabla \cdot \nabla \mathbf a  =   \mathop{\rm grad}  \mathop{\rm div}  \mathbf a ,\  \nabla
 +
[ \nabla , \mathbf a ]  =   \mathop{\rm div}  \mathop{\rm rot}  \mathbf a  = 0,
 +
$$
  
<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/h/h046/h046240/h04624027.png" /></td> </tr></table>
+
$$
 +
[ \nabla , [ \nabla , \mathbf a ] ]  =   \mathop{\rm rot} \
 +
\mathop{\rm rot}  \mathbf a ,\  \Delta \phi  = \nabla \cdot ( \nabla \phi )  =   \mathop{\rm div}  \mathop{\rm grad}  \phi .
 +
$$
  
 
The Hamilton operator was introduced by W. Hamilton [[#References|[1]]].
 
The Hamilton operator was introduced by W. Hamilton [[#References|[1]]].
Line 41: Line 145:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W.R. Hamilton,  "Lectures on quaternions" , Dublin  (1853)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W.R. Hamilton,  "Lectures on quaternions" , Dublin  (1853)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 19:43, 5 June 2020


nabla operator, $ \nabla $- operator, Hamiltonian

A symbolic first-order differential operator, used for the notation of one of the principal differential operations of vector analysis. In a rectangular Cartesian coordinate system $ x = ( x _ {1} \dots x _ {n} ) $ with unit vectors $ \mathbf e _ {1} \dots \mathbf e _ {n} $, the Hamilton operator has the form

$$ \nabla = \ \sum _ {j = 1 } ^ { n } \mathbf e _ {j} \frac \partial {\partial x _ {j} } . $$

The application of the Hamilton operator to a scalar function $ f $, which is understood as multiplication of the "vector" $ \nabla $ by the scalar $ f ( x) $, yields the gradient of $ f $:

$$ \mathop{\rm grad} f = \ \nabla f = \ \sum _ {j = 1 } ^ { n } \mathbf e _ {j} \frac{\partial f }{\partial x _ {j} } , $$

i.e. the vector with components $ ( \partial f / \partial x _ {1} \dots \partial f / \partial x _ {n} ) $.

The scalar product of $ \nabla $ with a field vector $ \mathbf a = ( a _ {1} \dots a _ {n} ) $ yields the divergence of $ \mathbf a $:

$$ \mathop{\rm div} \mathbf a = \ \nabla \mathbf a = \ \sum _ {j = 1 } ^ { n } \frac{\partial a _ {j} }{\partial x _ {j} } . $$

The vector product of $ \nabla $ with the vectors $ \mathbf a _ {j} = ( a _ {j1} \dots a _ {jn} ) $, $ j = 1 \dots n - 2 $, yields the curl (rotation, abbreviated by rot) of the fields $ \mathbf a _ {1} \dots \mathbf a _ {n-} 2 $, i.e. the vector

$$ [ \nabla , \mathbf a _ {1} \dots \mathbf a _ {n - 2 } ] = \ \left | \begin{array}{cccc} \mathbf e _ {1} &\mathbf e _ {2} &\dots &\mathbf e _ {n} \\ { \frac \partial {\partial x _ {1} } } &{ \frac \partial {\partial x _ {2} } } &\dots &{ \frac \partial {\partial x _ {n} } } \\ a _ {11} &a _ {12} &\dots &a _ {1n} \\ \cdot &\cdot &{} &\cdot \\ \cdot &\cdot &{} &\cdot \\ a _ {n - 2,1 } &a _ {n - 2,2 } &\dots &a _ {n - 2,n } \\ \end{array} \right | . $$

If $ n = 3 $,

$$ [ \nabla , \mathbf a ] = \nabla \times \mathbf a = \ \mathop{\rm rot} \mathbf a = \ \left ( \frac{\partial a _ {3} }{\partial x _ {2} } - \frac{\partial a _ {2} }{\partial x _ {3} } \right ) \mathbf e _ {1} + $$

$$ + \left ( \frac{\partial a _ {1} }{\partial x _ {3} } - \frac{\partial a _ {3} }{\partial x _ {1} } \right ) \mathbf e _ {2} + \left ( \frac{\partial a _ {2} }{\partial x _ {1} } - \frac{\partial a _ {1} }{\partial x _ {2} } \right ) \mathbf e _ {3} . $$

The scalar square of the Hamilton operator yields the Laplace operator:

$$ \Delta = \ \nabla \cdot \nabla = \ \sum _ {j = 1 } ^ { n } \frac{\partial ^ {2} }{\partial x _ {j} ^ {2} } . $$

The following relations are valid:

$$ [ \nabla , \nabla \phi ] = \ \mathop{\rm rot} \mathop{\rm grad} \phi = 0, $$

$$ \nabla \cdot \nabla \mathbf a = \mathop{\rm grad} \mathop{\rm div} \mathbf a ,\ \nabla [ \nabla , \mathbf a ] = \mathop{\rm div} \mathop{\rm rot} \mathbf a = 0, $$

$$ [ \nabla , [ \nabla , \mathbf a ] ] = \mathop{\rm rot} \ \mathop{\rm rot} \mathbf a ,\ \Delta \phi = \nabla \cdot ( \nabla \phi ) = \mathop{\rm div} \mathop{\rm grad} \phi . $$

The Hamilton operator was introduced by W. Hamilton [1].

References

[1] W.R. Hamilton, "Lectures on quaternions" , Dublin (1853)

Comments

See also Vector calculus.

References

[a1] D.E. Rutherford, "Vector mechanics" , Oliver & Boyd (1949)
[a2] T.M. Apostol, "Calculus" , 1–2 , Blaisdell (1964)
[a3] H. Holman, H. Rummler, "Alternierende Differentialformen" , B.I. Wissenschaftsverlag Mannheim (1972)
How to Cite This Entry:
Hamilton operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hamilton_operator&oldid=11494
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article