Second Derivatives and Laplace Operator

The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar or vector function yield second order expressions. Second-order expressions play a very important role in mathematics and physics. Some second order expressions include the divergence and curl of a gradient function, the divergence and curl of a curl function, and the gradient of a divergence function.

Consider a scalar function. The curl of its gradient can be written as follows:

For a vector function, the divergence of a curl can be expressed as follows:

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 can be expressed as follows:

The Laplacian is analogous to the second-order differentiation of the scalar quantities. It describes physical phenomena like electric potentials and diffusion equations for heat flow.

The divergence of gradient and the curl of a curl 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.