Del - Second Derivatives

Second Derivatives

When del operates on a scalar or vector, generally 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:

These are of interest principally because they are not always unique or independent of each other. As long as the functions are well-behaved, two of them are always zero:

Two of them are always equal:

The 3 remaining vector derivatives are related by the equation:

And one of them can even be expressed with the tensor product, if the functions are well-behaved:

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