gaussian in polar coordinates

Hi everyone,

i'm trying to express an 3-D gaussian distribution into spherical coordinates as part of a schoolwork. First of all, is there really any benefit in doing this? It is not obvious to me. Does it help in calculating the density estimate, etc.?

Here is the gaussian function of N-Dimensions in cartesian coordinates:

$\displaystyle

N(\vec{x}|\vec\mu, \Sigma) =

\frac{1}{{(2 \pi)}^{D/2} }

\frac{1}{| \Sigma | ^{1/2}}

e ^ {- \frac{1}{2} (\vec{x}-\vec\mu)^T\Sigma^{-1}(\vec{x}-\vec\mu) }

$

And I have to use spherical coordinates for 3-D gaussian:

$\displaystyle x_1 = r cos \phi sin \theta $

$\displaystyle x_2 = r sin \phi sin \theta $

$\displaystyle x_3 = r cos \theta $