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)