Dictionary:Del (∇)

{{#category_index:D|del (∇)}} The vector gradient operator. In rectangular coordinates,

$\nabla ={\textbf {i}}{\frac {\partial }{\partial x}}+{\textbf {j}}{\frac {\partial }{\partial y}}+{\textbf {k}}{\frac {\partial }{\partial z}}$ ,

where i, j, k are unit vectors in the x, y, z directions. $\nabla U$ is the gradient of the scalar field U.

The operator $\nabla ^{2}$, the Laplacian, appears frequently:

$\nabla ^{2}=\nabla \cdot \nabla ={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}$.

As an operator on a vector field V, $\nabla \cdot {\textbf {V}}$ is called the divergence, and $\nabla \times {\textbf {V}}$ is called the curl. Del is also called nabla and the vector operator. See also Figure C-14 for expressions using del in cylindrical and spherical coordinates.