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Triple integration

  • Thread starter big_gie
  • Start date
1. The problem statement, all variables and given/known data
Transform the equation from cartesians coordinates to spherical coordinates.


2. Relevant equations
[tex]\int_\infty\int_\infty\int_\infty
exp\left\{
\frac{-\left| \vec{x'}-\vec{x}_0 \right|^2}{2 \sigma}
\right\}
\frac{\left( \vec{x} - \vec{x'} \right)}{\left| \vec{x} - \vec{x'} \right|^{3}} d^3x'
[/tex]


3. The attempt at a solution
I'm confused by the [tex]\vec{x}[/tex], [tex]\vec{x'}[/tex] and [tex]\vec{x}_0[/tex]... I know it can be done: nothing depends on the angle here, so I should just get something depending on [tex]\vec{r}[/tex].

Thank you for any hints...
 

Defennder

Homework Helper
2,579
4
I'm equally confused by your integrand expression. It's supposed to be dxdydz isn't it? And what are the limits of your integration? And what does x' mean as both a scalar variable or vector function variable?
 
Hi Defennder, thanx for your reply.

Yes, [tex]d^3x'[/tex] is indeed [tex]dx'~dy'~dz'[/tex]: a volume element.

The integration is over infinity.

[tex]\vec{x'}[/tex] is the integration variable. It is a position vector [tex](x',y',z')[/tex].

Maybe I'll explain more the problem...

The integral is the electric field at position [tex]\vec{x}[/tex], caused by a charge distribution of gaussian shape:
[tex]\vec{E}\left(\vec{x}\right) =
k \int_{x'=-\infty}^{\infty} \int_{y'=-\infty}^{\infty} \int_{z'=-\infty}^{\infty}
\rho\left(\vec{x'}\right) \frac{\vec{x} - \vec{x'}}{\left| \vec{x} - \vec{x}'\right|^3} ~dx'~dy'~dz'[/tex]
[tex]
\rho\left(\vec{x}\right) & = & \rho_0 \exp\left(
-\frac{\left(\vec{x} - \vec{x_0}\right)^2}{2 \sigma^2}
\right)
[/tex]
where:
[tex]\vec{x}[/tex] is the position where the field is wanted;
[tex]\vec{x'}[/tex] is the integration variable;
[tex]\vec{x_0}[/tex] is the particle center;
[tex]\sigma[/tex] is the particle width.

I think I'll use the potential instead, for an easier integration:
[tex]\vec{E}\left(\vec{x}\right) = - \nabla \phi\left(\vec{x}\right)[/tex]
[tex]
\phi\left(\vec{x}\right) =
k \int_{x'=-\infty}^{\infty} \int_{y'=-\infty}^{\infty} \int_{z'=-\infty}^{\infty}
\rho\left(\vec{x'}\right) \frac{1}{\left| \vec{x} - \vec{x}'\right|} ~dx'~dy'~dz'
[/tex]

I've done a variable change for [tex]\vec{x} - \vec{x}'[/tex] but then I'm stuck in the gaussian...

Thanx for any hints.

(Sorry if any mistakes have slipped, I'm writting this from memory and it's getting late...)
 

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