I have a question regarding a change of variables inside a line integral. I attach an image with the configuration of problem and what I have done.
As I show in the image I know how to calculate the line integral if I directly use $\gamma (t) = (G \circ) (t)$. But what I really need to do is the following:
Take the integral over $\Gamma$ in the (x,y) space and rewrite it as a line integral over $\widehat{\Gamma}$ in the $(\alpha, \beta)$ space. Once this is done it will be enough to employ the parametrization of $\widehat{\Gamma}$ to calculate the integral, but for the moment I do not know how to do the first step.
I am actually interested in a more general configuration with a more general function $f(x,y)$ and a more general change of variables $G(\alpha,\beta)$. But I think for the moment it would be interesting to understand how the things work with the example I provide, with $f(x,y) = xy$ and $G(\alpha,\beta) = (2\alpha,\beta)$.
When I apply the change of variables in the integral on $\Gamma$, I do not know if I should employ the jacobian in the same way we do for area integrals. I have tried both introducing the jacobian and not doing it, and both give wrong answers... I think there is something a little bit more technical to be done that I do not understand.
If someone was able to help me I would be very grateful. Thank you in advance.
Let $\Gamma=(x(t),y(t))$ be a paramteric curve with ${t\in(t0,t1)}$
Then the line integral along $\Gamma$ is defined as following $$\oint_{\Gamma} f(x,y)ds = \int_{t_0}^{t1} f(x(t),y(t)) \frac{ds}{dt}dt= \int_{t_0}^{t1} f(x(t),y(t))\sqrt{x'(t)^2+y'(t)^2}dt$$ regardless of the parameterization.
Why the parameterization doesn't change anything? Simply because of $\frac{ds}{dt}$ which plays the role of jacobian.