Help on the proof related to the vectorial wave equation

1. Mar 8, 2015

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

2. Mar 9, 2015

RUber

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

3. Mar 10, 2015

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!

4. Mar 11, 2015

elgen

Worked it out. Thank you for the pointer.