I am struggling with taking the partial derivative of the following, specifically on the dependence of the variables in the partial derivative. This is the set up for a aerospace related problem.
$$r_{s} = \frac{a*(1-e^2)}{e*cos(\nu)+1}$$
$$Range = \sqrt{r_{s}^{2} + RE^2 + RE*r_{s}*cos(\phi)}$$
Where $\phi(\nu) = \cos^{-1}\({\cos{\theta(\nu)}\cos{\psi}}\)$ and $\theta = \nu - \lambda$.
My question now essentially arises when having to take the partial of Range. It is clear that when doing $\frac{\partial Range}{\partial \nu}$ that the chain rule applies to $\frac{\partial \phi}{\partial \nu}$, However if I want to do $\frac{\partial Range}{\partial \phi}$ does the chain rule apply to $\frac{\partial \nu}{\partial \phi}$? Based on how I declared the variables, $\phi$ is defined using $\nu$ but $\nu$ is independent of $\phi$.
So is $\frac{\partial \nu}{\partial \phi}=0$?.