Hamilton operator
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) |
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