Nabla operator¶

Definition¶

The nabla operator is defined by the following expression,

(59)¶\[\vec{\nabla}=\frac{\partial}{\partial x}\vec{i}+\frac{\partial}{\partial y}\vec{j}+\frac{\partial}{\partial z}\vec{k}\]

This is an operator, meaning that, depending on how it is combined with other object it acts in a different way. This is really a nice trick to do quite different things using always well-known algebra.

Uses¶

Gradient¶

The gradient is obtained applying the nabla operator on a scalar field leading to a vector field.

(60)¶\[$\vec{\nabla}f(x,y,z)\]

Divergence¶

The divergence is obtained applying the nabla operator on a vector field as a dot product leading to a scalar field.

(62)¶\[\vec{\nabla}\cdot\vec{f}(x,y,z)\]

Rotacional¶

The rotational is obtained applying the nabla operator on a vector field as a cross product leading to a vector field.

(62)¶\[\vec{\nabla}\times\vec{f}(x,y,z)\]

Problems and solved examples¶

Partial derivatives Gradient

Física General