2.11 The Laplacian Operator

All of the differential operations discussed so far pertain to first-order differential operators. One second-order differential operator that occurs frequently in the study of field theory is called the Laplacian operator, symbolically written as ? ². It is defined as the divergence of a gradient of a scalar function. Simply put, if f( x, y, z) is a continuously differentiable scalar function, the Laplacian of f( x, y, z) is

We can write the divergence of a scalar function f in the rectangular coordinate system as

which yields

From (2.105) it is evident that the Laplacian of a scalar function is a scalar and involves second-order partial differentiation of the function. By simple transformations, we can express the Laplacian of a scalar function in cylindrical coordinates as

A similar transformation from rectangular to spherical coordinates will yield the Laplacian of a scalar in spherical coordinates as

A scalar function is said to be a harmonic function if its Laplacian is zero. That is,

This equation is routinely referred to as Laplace s equation.

In our discussion of electromagnetic fields we will also encounter expressions of the form ? ² , where is a vector field. We call such an expression the Laplacian of a vector field and define it as

In the Cartesian coordinate system, equation (2.109) becomes

where

is the Laplacian operator. The Laplacian of a vector field is zero if and only if the...