Can Laplacian and Curl Operators Be Interchanged?

[Karthiksrao]

Hi,

During the description of vector spherical harmonics, where N = curl of M , I came across the following :

Laplacian of N = Laplacian of (Curl of M) = Curl of (Laplacian of M)

How do we know that these operators can be interchanged ? What is the general rule for such interchanges ?

Thanks

 

[Pengwuino]

The Laplacian is a scalar operator. It can move past other derivatives

 

[dextercioby]

To understand where this mambo-jumbo with vectors/scalars and differential operators all comes from, you need to know how to use tensor notation. Specifically, let's assume you're working in the cartesian system of coordinates.

Then

$$N_i = \epsilon_{ijk} \partial_j M_k$$ and the Laplacian should act like

$$\partial_m \partial_m N_i = \epsilon_{ijk} \partial_m \partial_m \partial_j M_k$$

Now, M's components are well behaved functions and you can assume interchanging the 3 differential operators acting on them.

You'll find easily that what your text is asserting is, well, true...