I learned a technic to convert a usual integral (that is, integrator is $x^n$) to polar-coordinate integral.
To do this process formally, I think one should know surface measure and Fubini's theorem and these theorems are not really basic. Moreover I don't know whether this process can be formalized for Riemann integral and even though this can be, I think it's not gonna be easier.
However, in very elementary calculus book, authors introduce polar-coordinates and use them very frequently, and I'm very uncomfortable with this.
Just like Integral, what is the formal definition of polar-coordinate partial derivative and how is this related to ordinary partial derivative
To be specific, here is an example.
Let $f:\mathbb{C}\rightarrow \mathbb{C}$ be a function.
What does it exactly mean by $\frac{\partial f}{\partial \theta}(z)$?
Does this mean $\lim_{{\theta\to Arg(z)}\bidwedge \{theta\in (-\pi,\pi]} \frac{f(|z|e^{i \theta)}) - f(|z|^{i Arg(z)})}{\theta - Arg(z)}$?