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After correspondence with the author of those notes, I arrived at the following conclusion. When we do $\frac{\partial f}{\partial r}$, we require that all other (other than $r$) variables which f is dependent on, are held constant. That is: $\theta, \phi, p_r, p_{\theta}, p_{\phi}$ are constants. Note that: $(p_r, p_{\theta}, p_{\phi})=(\dot{r}, r^2 \sin^2 \...

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