Is My Interpretation of the 2nd Order Derivative Correct?

[test1234]

Hi there, I'm kind of rusty on some stuff, so hope someone can help enlighten me.

I have an expression
E(r,w-w0)=F(x,y) A(z,w-w0) \exp[i\beta_0 z]

I need to substitute this into the Helmholtz equation and solve using separation of variables. However, I'm getting problems simplifying it to a form with can be separated... I reckon the problem lies with my understanding of the 2nd derivative, especially with more variables coming into play.

From the Helmholtz equation,
\nabla^2 E+\epsilon (w) k_0^{\phantom{0}2} E=0

Working out
\nabla^2 E =\nabla(\nabla E)
=\nabla(A\exp[i\beta_0 z] \frac{\partial F}{\partial x}+A\exp[i\beta_0 z] \frac{\partial F}{\partial y}+ FA (i \beta_0 \exp[i\beta_0 z])+F \exp[i\beta_0 z] \frac{\partial A}{\partial z})<br />
<br /> =A \exp[i\beta_0 z] \frac{\partial^2 F}{\partial x^2}+i \beta_0 A \exp[i\beta_0 z] \frac{\partial F}{\partial x}+ \exp[i\beta_0 z] \frac{\partial A}{\partial z} \frac{\partial F}{\partial x}
<br /> + A \exp[i\beta_0 z] \frac{\partial^2 F}{\partial y^2}+i \beta_0 A \exp[i\beta_0 z] \frac{\partial F}{\partial y}+\exp[i\beta_0 z] \frac{\partial A}{\partial z} \frac{\partial F}{\partial y}<br />
<br /> +\frac{\partial F}{\partial x} \exp[i\beta_0 z] \frac{\partial A}{\partial z}+ \frac{\partial F}{\partial y} \exp[i\beta_0 z] \frac{\partial A}{\partial z}+ F A (i \beta_0)^2 \exp[i\beta_0 z]<br />
<br /> +i \beta_0 F \exp[i\beta_0 z] \frac{\partial A}{\partial z}+F \frac{\partial A}{\partial z} (i \beta_0) \exp[i\beta_0 z]+F \exp[i\beta_0 z] \frac{\partial^2 A}{\partial z^2}<br />
Which gives
<br /> =\exp[i\beta_0 z][A \frac{\partial^2 F}{\partial x^2}+2i \beta_0 A \frac{\partial F}{\partial x}+ 2 \frac{\partial A}{\partial z} \frac{\partial F}{\partial x}+A \frac{\partial^2 F}{\partial y^2}+2i \beta_0 A \frac{\partial F}{\partial y}+ 2 \frac{\partial A}{\partial z} \frac{\partial F}{\partial y}]<br />


It seems like the terms 2 \frac{\partial A}{\partial z} \frac{\partial F}{\partial x} and 2 \frac{\partial A}{\partial z} \frac{\partial F}{\partial y} need to vanish...

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Or have I done the derivative wrongly? Should it be the following instead?
<br /> \nabla^2 E =\nabla_x ^{\phantom{0}2}E+\nabla_y ^{\phantom{0}2}E+\nabla_z ^{\phantom{0}2}E<br />

where
<br /> \nabla_x ^{\phantom{0}2}E = A \exp[i\beta_0 z] \frac{\partial^2 F}{\partial x^2}<br />

<br /> \nabla_y ^{\phantom{0}2}E = A \exp[i\beta_0 z] \frac{\partial^2 F}{\partial x^y}<br />


<br /> \nabla_z^{\phantom{0}2}E = \nabla_z [FA (i \beta_0) \exp[i\beta_0 z]+ F \exp[i\beta_0 z] \frac{\partial A}{\partial z}]<br />
<br /> =[F (i \beta_0) \exp[i\beta_0 z] \frac{\partial A}{\partial z}+FA (i \beta_0)^2 \exp[i\beta_0 z]+<br /> F \exp[i\beta_0 z] \frac{\partial^2 A}{\partial z^2}]+ F \frac{\partial A}{\partial z} (i \beta_0) \exp[i\beta_0 z]<br />

Thanks in advance!

 

[SteamKing]

The Laplacian operator ∇^{2} is defined as ∇\cdot∇, or in cartesian components

∇^{2}f = ∂^{2}f/∂x^{2} + ∂^{2}f/∂y^{2} + ∂^{2}f/∂z^{2}

http://en.wikipedia.org/wiki/Del

 

[test1234]

Thanks SteamKing!

I suppose my second intepretation should be the correct one instead.