I am following these notes: Dynamics and Astrophysics of Galaxies.
After equation 6.37, we have: \begin{equation*} p_r\,\frac{\partial f}{\partial r} + \frac{p_\theta}{r^2}\,\frac{\partial f}{\partial \theta} + \frac{p_\phi}{r^2\,\sin^2\theta}\,\frac{\partial f}{\partial \phi} -\left(\frac{\mathrm{d} \Phi}{\mathrm{d} r}-\frac{p_\theta^2}{r^3}-\frac{p_\phi^2}{r^3\,\sin^2\theta}\right)\,\frac{\partial f}{\partial p_r} +\frac{p_\phi^2\,\cos\theta}{r^2\,\sin^3\theta}\,\frac{\partial f}{\partial p_\theta} = 0\,.\\ \end{equation*}
This is the Collisionless Boltzmann Equation in Spherical Polar Coordinates.
Then
We now multiply this by $p_r$ and integrate over all $(p_r,p_{\phi},p_{\theta})$ using that $\mathrm{d}p_r\,\mathrm{d}p_\phi\,\mathrm{d}p_\theta = r^2\,\sin\theta\,\mathrm{d}v_r\, \mathrm{d}v_\phi\,\mathrm{d}v_\theta$ and using partial integration to deal with the derivatives of f with respect to the momenta
\begin{align}\label{eq-spher-jeans-penult} \frac{\partial (r^2\,\sin\theta\,\nu\,\overline{v^2_r})}{\partial r} + \frac{\partial (\sin\theta\,\nu\,\overline{v_r\,v_\theta})}{\partial \theta} & + \frac{\partial (\nu\,\overline{v_r\,v_\phi}/\sin\theta)}{\partial \phi}\\ & +r^2\,\sin\theta\,\nu\,\left(\frac{\mathrm{d} \Phi}{\mathrm{d} r}-\frac{\overline{v_\theta^2}}{r}-\frac{\overline{v_\phi^2}}{r}\right) = 0\nonumber\,. \end{align}
I would like to arrive to this result myself. Multiplying the SPC CBE by $p_r$ & going ahead with the suggested volume element, considering the first term in the above equation, not being worried about the integration limits gives us:
\begin{equation} \int p_r \frac{\partial f}{\partial r} p_r r^2 \sin\theta \mathrm{d}v_r \mathrm{d}v_{\theta} \mathrm{d}v_{\phi} =\int v_r^2 \frac{\partial f}{\partial r} r^2 \sin \theta \mathrm{d}v_r \mathrm{d}v_{\theta} \mathrm{d}v_{\phi} =r^2 \sin \theta \int v_r^2 \frac{\partial f}{\partial r} \mathrm{d}v_r \mathrm{d}v_{\theta} \mathrm{d}v_{\phi} \end{equation}
Using 6.32 from the mentioned notes,
\begin{equation} \frac{\partial (r^2\,\sin\theta\,\nu\,\overline{v^2_r})}{\partial r} = \frac{\partial}{\partial r} \left( r^2 \sin\theta \int v^2_r f \mathrm{d}v_r \mathrm{d}v_{\theta} \mathrm{d}v_{\phi} \right) \end{equation}
Which is not equal to what I have found just a line above.
What am I doing wrong?
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