Represent $g \sim N(0,I_n)$in polar form as $g=r \theta$ where $r = \|g\|_2$ is the length and $\theta = \frac{g}{\|g\|_2} $ is the direction
prove that $r$ and $\theta$ are independent ?
prove that $\theta$ is uniformly distributed on sphere $S^{n-1}$
for first one : The only things I know how to do is to show the product pdf of both of $\theta$ and $r$ is same as n-dimeinal pdf of of standard Gaussian vector ? but how to find pdf of $\|g\|_2$? I have some problem in trasformation of random variable in this case . since the transformationare not bijective , is any simple way to do that?
Big hint: The joint density of $G = (X, Y)$ is $f(x,y) \, dx \, dy = \frac{1}{2\pi} e^{-(x^2+y^2)/2} \, dx \, dy$. By performing a change of variables to polar coordinates $R^2 = X^2 + Y^2$ and $\Theta = \arctan(Y/X)$, we have $$f(r, \theta) \, dr \, d\theta = \frac{1}{2\pi} e^{-r^2/2} r \, dr \, d\theta,$$ where the extra factor of $r$ comes from the Jacobian when performing the change of variables.
Hint: If you know the joint density can be written as $f(r, \theta) = g(r) h(\theta)$ for some densities $g$ and $h$, then $R$ and $\Theta$ are independent.