Tweede derivaten en Laplace-operator

The first-order operators using the del operator includes the gradient, divergence, and curl.

Certain combinations of first-order operators on a scalar or a vector function yields second-order expressions.

The second order derivatives include: the divergence and curl of a gradient function, the divergence and curl of a curl function, and the gradient of a divergence function.

The curl of a gradient function and the divergence of a curl function are always zero.

The divergence of the gradient of a scalar function gives the scalar Laplace operator, or Laplacian. A Laplacian is analogous to the second-order derivative of the scalar quantities.

When the gradient of a scalar function is expressed in cylindrical and spherical coordinates, its Laplacian in cylindrical and spherical coordinates is obtained.

The gradient of a divergence function and the curl of a curl function are mathematical constructs. Lagrange's vector cross-product identity formula relates both to a vector Laplacian.

The vector Laplacian is obtained by directly applying the scalar Laplacian to each of the scalar components of a vector.