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
The scalar square of the Hamilton operator yields the Laplace operator:
The following relations are valid:
The Hamilton operator was introduced by W. Hamilton .
|||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)|
Hamilton operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hamilton_operator&oldid=11494