Hamilton operator

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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

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 :

i.e. the vector with components .

The scalar product of with a field vector yields the divergence of :

The vector product of with the vectors , , yields the curl (rotation, abbreviated by rot) of the fields , i.e. the vector

If ,

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

The following relations are valid:

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


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


See also Vector calculus.


[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:
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article