Proving the scalar wave equation from vectorial curl identity

[elgen]

Dear forum users,

I need some help on the following proof that appears in a book (pp. 84 in Bohren' Absorption and Scattering of light by Small Particles). This is no a home work problem.

The problem statement:

$\vec{M} = \nabla\times\vec{c}\psi$, where $\vec{c}$ is some constant vector and $\psi$ is a scalar function, then if $-\nabla\times\nabla\times\vec{M}+k^2\vec{M}=0$ where $k$ is the wave number, then prove that $\nabla^2 \psi + k^2\psi=0$.

I could prove for the cases by using a curve-linear coordinate system, etc. rectangular, cylindrical, etc. I am seeking a general proof. I suspect that there is some vectorial identity applicable here.

Thank you for the attention.Elgen

 

[RUber]

What happens if you use $\nabla \times c \psi = c \nabla \times \psi +\nabla c \times \psi$ and $\nabla \times \nabla \times A = \nabla (\nabla \cdot A) - \nabla^2 A$?

 

[vanhees71]

The (Cartesian) Ricci Calculus is your friend here. But isn't this more homework like? So I'd say, this thread should be moved to the homework section of these forums!

 

[elgen]

Worked it out. Thank you for the pointer.