# 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].

#### References

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