Del operator

The del operator (∇) is an operator commonly used in vector calculus to find derivatives in higher dimensions. When applied to a function of one independent variable, it yields the derivative. For multidimensional scalar functions, it yields the gradient. If either dotted or crossed with a vector field, it produces divergence or curl, respectively, which are the vector equivalents of differentiation.

$\begin{align} &\nabla=\begin{bmatrix} \dfrac{\part}{\part x}\\\dfrac{\part}{\part y}\\\dfrac{\part}{\part z}\end{bmatrix}\\ &\text{grad}(f)=\nabla f\\ &\text{div}(\mathbf{F})=\nabla\cdot\mathbf{F}\\ &\text{curl}(\mathbf{F})=\nabla\times\mathbf{F} \end{align}$

There are six ways del can be used to compute second derivatives of multivariable functions.

• $\nabla\cdot\nabla f=\nabla^2f$ The divergence of the gradient, also know as the Laplacian
• $\nabla\cdot\nabla\mathbf{F}=\nabla^2\mathbf{F}$ The vector Laplacian, equal to the Laplacian of each component of the vector
• $\nabla\times\nabla f=0$ The curl of the gradient, always equal to 0 (see irrotational vector field)
• $\nabla(\nabla\cdot\mathbf{F})$ The gradient of the divergence
• $\nabla\cdot(\nabla\times\mathbf{F})=0$ The divergence of curl, always equal to 0 (see incompressible vector field)
• $\nabla\times(\nabla\times\mathbf{F})=\nabla(\nabla\cdot\mathbf{F})-\nabla^2\mathbf{F}$ The curl of the curl