Laplacian operating on a vector/scalar product

Can someone work through the following for me, or show me a derivation, I'm getting in a right muddle:

- \nabla^2 (\bold{E_0} exp(i \bold{k.x} - iwt)) = \bold{k.k}( \bold{E_0} exp(i \bold{k.x} - iwt))

For a start, is x here supposed to be some position vector with components

(x1, x2, x3)

or is it

\bold{x} = x\bold{i}

?

Do I then need to use some identity for the laplacian of a vector/scalar product, and then work each of them out or what?

x looks like a general position vector. You should be able to do this by rewriting everything in components...

Yep. x is a general position vector.

\nabla^2 = \frac{\partial}{\partial x_j}\frac{\partial}{\partial x_j}

for cartesians.

I tried it in component form and got nowhere - I must have been doing it wrong?

So in component form:

\frac{\partial}{\partial x_i}\frac{\partial}{\partial x_i}(E^j e^{(ik^kx^k - iwt)})

so it'll be

\frac{\partial}{\partial x_i} \frac{\partial E^j}{\partial x_i}(e^{(ik^kx^k - iwt)}) + E^j \frac{\partial}{\partial x_i} \frac{\partial}{\partial x_i}(e^{(ik^kx^k - iwt)})

whatever the latter differential turns out to be? If so I'll get calculating.

Barny

I tried it in component form and got nowhere - I must have been doing it wrong?

So in component form:

\frac{\partial}{\partial x_i}\frac{\partial}{\partial x_i}(E^j e^{(ik^kx^k - iwt)})

so it'll be

\frac{\partial}{\partial x_i} \frac{\partial E^j}{\partial x_i}(e^{(ik^kx^k - iwt)}) + E^j \frac{\partial}{\partial x_i} \frac{\partial}{\partial x_i}(e^{(ik^kx^k - iwt)})

whatever the latter differential turns out to be? If so I'll get calculating.

E_j is constant. so it's just the latter half of that expression

Oh dear. My notes even say "real, constant three vectors" :frown:

Getting back into studying is hard. Just as I was thinking it seemed like a lot of algebra. Thank you yusufu

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