Hamilton operator

nabla operator, $\nabla$- 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 $x = ( x _ {1} \dots x _ {n} )$ with unit vectors $\mathbf e _ {1} \dots \mathbf e _ {n}$, the Hamilton operator has the form

$$\nabla = \ \sum _ {j = 1 } ^ { n } \mathbf e _ {j} \frac \partial {\partial x _ {j} } .$$

The application of the Hamilton operator to a scalar function $f$, which is understood as multiplication of the "vector" $\nabla$ by the scalar $f ( x)$, yields the gradient of $f$:

$$\mathop{\rm grad} f = \ \nabla f = \ \sum _ {j = 1 } ^ { n } \mathbf e _ {j} \frac{\partial f }{\partial x _ {j} } ,$$

i.e. the vector with components $( \partial f / \partial x _ {1} \dots \partial f / \partial x _ {n} )$.

The scalar product of $\nabla$ with a field vector $\mathbf a = ( a _ {1} \dots a _ {n} )$ yields the divergence of $\mathbf a$:

$$\mathop{\rm div} \mathbf a = \ \nabla \mathbf a = \ \sum _ {j = 1 } ^ { n } \frac{\partial a _ {j} }{\partial x _ {j} } .$$

The vector product of $\nabla$ with the vectors $\mathbf a _ {j} = ( a _ {j1} \dots a _ {jn} )$, $j = 1 \dots n - 2$, yields the curl (rotation, abbreviated by rot) of the fields $\mathbf a _ {1} \dots \mathbf a _ {n-} 2$, i.e. the vector

$$[ \nabla , \mathbf a _ {1} \dots \mathbf a _ {n - 2 } ] = \ \left | \begin{array}{cccc} \mathbf e _ {1} &\mathbf e _ {2} &\dots &\mathbf e _ {n} \\ { \frac \partial {\partial x _ {1} } } &{ \frac \partial {\partial x _ {2} } } &\dots &{ \frac \partial {\partial x _ {n} } } \\ a _ {11} &a _ {12} &\dots &a _ {1n} \\ \cdot &\cdot &{} &\cdot \\ \cdot &\cdot &{} &\cdot \\ a _ {n - 2,1 } &a _ {n - 2,2 } &\dots &a _ {n - 2,n } \\ \end{array} \right | .$$

If $n = 3$,

$$[ \nabla , \mathbf a ] = \nabla \times \mathbf a = \ \mathop{\rm rot} \mathbf a = \ \left ( \frac{\partial a _ {3} }{\partial x _ {2} } - \frac{\partial a _ {2} }{\partial x _ {3} } \right ) \mathbf e _ {1} +$$

$$+ \left ( \frac{\partial a _ {1} }{\partial x _ {3} } - \frac{\partial a _ {3} }{\partial x _ {1} } \right ) \mathbf e _ {2} + \left ( \frac{\partial a _ {2} }{\partial x _ {1} } - \frac{\partial a _ {1} }{\partial x _ {2} } \right ) \mathbf e _ {3} .$$

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

$$\Delta = \ \nabla \cdot \nabla = \ \sum _ {j = 1 } ^ { n } \frac{\partial ^ {2} }{\partial x _ {j} ^ {2} } .$$

The following relations are valid:

$$[ \nabla , \nabla \phi ] = \ \mathop{\rm rot} \mathop{\rm grad} \phi = 0,$$

$$\nabla \cdot \nabla \mathbf a = \mathop{\rm grad} \mathop{\rm div} \mathbf a ,\ \nabla [ \nabla , \mathbf a ] = \mathop{\rm div} \mathop{\rm rot} \mathbf a = 0,$$

$$[ \nabla , [ \nabla , \mathbf a ] ] = \mathop{\rm rot} \ \mathop{\rm rot} \mathbf a ,\ \Delta \phi = \nabla \cdot ( \nabla \phi ) = \mathop{\rm div} \mathop{\rm grad} \phi .$$

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

References

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