I am having difficulties with the following formula in order to compute the arc length of a function.
$$L = \int_{\theta_1}^{\theta_2} \sqrt{\left(\frac{\mathrm{d}r}{\mathrm{d}\theta}\right)^2+r^2} \ \mathrm{d}\theta$$
My problem is that I have a function $r(\theta,\phi)$ that is defined on $r:\mathbb{R}^2 \rightarrow \mathbb{R}$, and I need to compute the arc length between $\theta_1$ and $\theta_2$ with $\phi$ being fixed. In other words, can I compute $L$ by computing $\mathrm{d}r / \mathrm{d}\theta$ in the same way as I would treat $\partial r / \partial\theta$ ?