Dictionary:Del (∇)

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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. </translate>