# Dictionary:Del (∇)

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Other languages:
English • ‎español

The vector gradient operator. In rectangular coordinates,

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

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

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

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

As an operator on a vector field V, ${\displaystyle \nabla \cdot {\textbf {V}}}$ is called the divergence, and ${\displaystyle \nabla \times {\textbf {V}}}$ 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.