# 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 .

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