Background
I want to define a velocity distribution function in the 3D Euclidean space ($v_x$, $v_y$, $v_z$), where any vector $\vec{v}$ (defining a velocity vector in the physical space) is represented by a point.
My distribution function follows an exponentially-modified Gaussian (in energy) along one specific direction, defined as the unit vector $\hat{v_B}$.
Ultimately, I will need to define this distribution function in spherical coordinates ($E$, $\theta$, $\phi$).
Solution attempt
I defined my distribution function as:
$f(\vec{v}) = \delta(\hat{v} - \hat{v}_B) \times \dfrac{\lambda}{2}e^{\dfrac{\lambda}{2}(2\mu+\lambda\sigma^2-2E)}erfc(E)$
The second part of the equation defines the expo-modified gaussian in energy (E=0.5mv^2). Is this function mathematically sound, first of all?
Then, to express $f(\vec{v})$ wrt. ($E$, $\theta$, $\phi$), I first converted $\delta(\hat{v} - \hat{v}_B)$ as follows:
$v_x = v\sin(\theta)\cos(\phi)$
$v_y = v\sin(\theta)\cos(\phi)$
$v_z = v\cos(\theta)$
Giving:
$\delta(\hat{v} - \hat{v}_B) = \dfrac{1}{\sin(\theta)} \delta(\theta - \theta_B) \delta(\phi - \phi_B)$
The expo-modified gaussian is already defined in E, so we have:
$f(E, \theta, \phi) = \dfrac{1}{\sin(\theta)} \delta(\theta - \theta_B) \delta(\phi - \phi_B) \times \dfrac{\lambda}{2}e^{\dfrac{\lambda}{2}(2\mu+\lambda\sigma^2-2E)}erfc(E)$
So far, this is the furthest I have been. One other way to define my 3D delta dirac would be: $\delta(\vec{v} - \hat{v}_Bv)$.
Do you see any issues with the derivations above?
Thank you for the help!!!