# 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

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].

#### 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) |

**How to Cite This Entry:**

Hamilton operator.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Hamilton_operator&oldid=11494