Laplace operator plays a vital role in many outstanding problems, such as Helmholtz’s equations, Allen-Cahn’s equations, sine-Gordon equations, and so forth. In Cartesian coordinates, it is denoted by and has the following form.
However, its form in spherical coordinates is
We aim to prove this formula, means that converting the operator from Cartesian to spherical coordinates is in order.
As usual, we have the following transformation.
where are called radius, inclination and azimuth, respectively.
It follows that
Combining (1) and (2), we first have
Moreover, the derivatives of with respect to are
Simultaneously, the derivatives of are simply
Observing chain rule is now necessary. For ,
By substituting (3)-(4)-(5) into (6), we then have
In brief, we shall obtain from (7)-(8)-(9) that
which gives the desired result.
It is so surprising that the final result is so short after directly having extremely heavy computation. It requires meticulosity, diligence and none of scare when confront with tremendous computations. But do not try to pay more attention to this, the important thing we finally obtain is the formula of Laplace operator in spherical coordinates. From this formula, we also would like to introduce the more general case which is known as the Laplace-Beltrami operator. However, for further posts, we still aim to focus on many issues about Laplace operator in spherical coordinates on account of a wide range of applications.