Editing Del (section)

==Second derivatives==
[[File:DCG chart.svg|thumb|DCG chart:

A simple chart depicting all rules pertaining to second derivatives.
D, C, G, L and CC stand for divergence, curl, gradient, Laplacian and curl of curl, respectively.

Arrows indicate existence of second derivatives. Blue circle in the middle represents curl of curl, whereas the other two red circles (dashed) mean that DD and GG do not exist. 
]]
When del operates on a scalar or vector, either a scalar or vector is returned. Because of the diversity of vector products (scalar, dot, cross) one application of del already gives rise to three major derivatives: the gradient (scalar product), divergence (dot product), and curl (cross product). Applying these three sorts of derivatives again to each other gives five possible second derivatives, for a scalar field ''f'' or a vector field '''''v'''''; the use of the scalar [[Laplacian]] and [[vector Laplacian]] gives two more:
: <math>\begin{align}
        \operatorname{div}(\operatorname{grad}f) &= \nabla \cdot (\nabla f) = \nabla^2 f \\
       \operatorname{curl}(\operatorname{grad}f) &= \nabla \times (\nabla f) \\
   \operatorname{grad}(\operatorname{div}\mathbf v) &= \nabla (\nabla \cdot \mathbf v) \\
   \operatorname{div}(\operatorname{curl}\mathbf v) &= \nabla \cdot (\nabla \times \mathbf v) \\
  \operatorname{curl}(\operatorname{curl}\mathbf v) &= \nabla \times (\nabla \times \mathbf v) \\
                                        \Delta f &= \nabla^2 f \\
                                   \Delta \mathbf v &= \nabla^2 \mathbf v
\end{align}</math>

These are of interest principally because they are not always unique or independent of each other.  As long as the functions are well-behaved (<math> C^\infty</math> in most cases), two of them are always zero:
: <math>\begin{align}
      \operatorname{curl}(\operatorname{grad}f) &= \nabla \times (\nabla f) = 0 \\
  \operatorname{div}(\operatorname{curl}\mathbf v) &= \nabla \cdot (\nabla \times \mathbf v) = 0
\end{align}</math>

Two of them are always equal:
: <math> \operatorname{div}(\operatorname{grad}f) = \nabla \cdot (\nabla f) = \nabla^2 f = \Delta f </math>

The 3 remaining vector derivatives are related by the equation:
:<math>\nabla \times \left(\nabla \times \mathbf v\right) = \nabla (\nabla \cdot \mathbf v) - \nabla^2 \mathbf{v}</math>

And one of them can even be expressed with the tensor product, if the functions are well-behaved:
: <math>\nabla (\nabla \cdot \mathbf v) = \nabla \cdot (\mathbf v \otimes \nabla )</math>