Just like the title says, How does the ellipsis with equation $$x^2+2y^2=2$$ becomes represented as $$x=\sqrt{2}\cos\theta; y= \sin(\theta)$$ in polar coordinates?
can someone help me to understand the translation between cartesian to polar here?
This is for evaluating a double integral, btw. But I can't understand on my own how to get that transformation.
The idea of the transformation here is based on polar coordinates but actually is called a parametrisation (as stated in the comments). Precisely, you wish to find values of $x$ and $y$ such that the above equation is true. In your case, plugging in $x=\sqrt{2}\cos\theta$ and $y=\sin\theta$ yields $$(\sqrt{2}\cos\theta)^2+2\sin^2\theta=2\cos^2\theta+2\sin^2\theta=2,$$ since $\cos^2\theta+\sin^2\theta=1$. Then, instead of evaluating a double integral over $x$ and $y$, you just use your transformation and evaluate it over $r$ and $\theta$, of course without forgetting the Jacobian.