$\newcommand{\atan}{\operatorname{atan2}}$On the one-form page on Wikipedia (and other sources), the change in angle formula $\theta=\atan(y,x)$ (defined up to an additive constant) leads to the total derivative: $$ d\theta = -\frac{y}{x^2+y^2} \, dx + \frac{x}{x^2+y^2} \, dy. $$
This derivative is continuously defined except at the origin. Integrating this quantity over a closed loop gives winding number times $2\pi$.
- Now, I have some arbitrary function $f(\theta)$ from physics that I want to integrate over a closed loop. I can perform the integral $\oint f(\theta) \, d\theta$ with no issue. Am I right in saying that I can rewrite this integration in terms of $x,y$ as follows:
$$ \oint f(\theta) \, d\theta = \oint \left[-f(\atan(y,x))\frac{y}{x^2+y^2} \, dx + f(\atan(y,x))\frac{x}{x^2+y^2} \, dy\right] \text{?} $$
- Now, the arbitrary function $f(\theta)$ above happens contain $\theta$ derivatives of a function $g(\theta)$: $$f(\theta)=\frac d {d\theta} g(\theta)+\frac{d^2}{d\theta^2} g(\theta).$$ How do I rewrite the derivative in terms of derivatives of $x,y$? I think, by the chain rule, it should be: $$ \frac d {d\theta} = \frac\partial {\partial x} \frac{dx}{d\theta} + \frac\partial{\partial y}\frac{dx}{d\theta}. $$
But is there no way to write this without having the $\frac{dx}{d\theta}$ and $\frac{dy}{d\theta}$ terms directly? At least in terms of different notation? For instance, I was thinking that I could define a 'unit tangent vector' along the loop and allow it to contain these derivatives/speeds.
- Can I write a proper 2-form using the 1-form product $f(\theta) \, d\theta$, only in terms of $x,y$? I was thinking I could do: $$ \frac d {dx} \left[f(\atan(y,x))\frac{x}{x^2+y^2}\right] - \frac d {dy} \left[-f(\atan(y,x))\frac{y}{x^2+y^2}\right] \, dx \wedge dy, $$
but I am not sure about this. Since I want to understand this in broad generality, I want to do this without using specific examples of $f,g$. Thank you.