I have a function defined over the spherical coordinates $f(r,\theta,\phi)$. A want to perform a path integral along a line parallel to the $y$-axis. Hence, the values of x and z are fixed along the path, only $y$ varies. Let's call the fixed values of $x=x_0$ and $z=z_0$
$$\int_{y_0}^{y_1} f(r(x_0,y,z_0),\theta(x_0,y,z_0),\phi(x_0,y,z_0))dy$$
I want an expression of this integral as a function of $(r,\theta,\phi,dr,d\theta,d\phi)$ only, not of $y$ and $dy$.
I am unsure how to procede, as it is easy to integrate this over $dy$ as the function is already expressed in $y$, but to integrate this over $(r,\theta,\phi)$ is harder. Am I supposed to take the total differential: $$dy=\frac{\partial y}{\partial r}dr +\frac{\partial y}{\partial \theta}d\theta +\frac{\partial y}{\partial \phi}d\phi $$
and integrate over this? It somewhat feels wrong and I am unsure what to do. For instance, if I were to try this, I would get a sum of $3$ integrals, with the first one being:
$$\int_{r_0}^{r_1} f(r,\theta,\phi)\frac{\partial y}{\partial r}dr$$
which doesn't seem to make much sense. For instance, if $y_1=-y_0$, then $r_1=r_0$, so the above integral should be zero? And how does the fact that my $x$ and $z$ coordinates are fixed come into play when trying to do this integral?
Thanks for the help!