Hamilton operator
nabla operator, -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 with unit vectors
, the Hamilton operator has the form
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The application of the Hamilton operator to a scalar function , which is understood as multiplication of the "vector"
by the scalar
, yields the gradient of
:
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i.e. the vector with components .
The scalar product of with a field vector
yields the divergence of
:
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The vector product of with the vectors
,
, yields the curl (rotation, abbreviated by rot) of the fields
, i.e. the vector
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If ,
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The scalar square of the Hamilton operator yields the Laplace operator:
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The following relations are valid:
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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) |
Hamilton operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hamilton_operator&oldid=11494