So I was scrolling through the homepage of Youtube when I came across this $20$ second video by Abdallah El Daly which shows the steps that would be taken to solve the Gaussian Integral $$\int_{-\infty}^\infty e^{-x^2}$$using polar coordinates to solve. However, $7$ seconds into the video (around $4$ seconds after it starts showing the solution in fast speed), it shows these steps:$$\left(\iint_{\mathbb{R}^+\times[0,2\pi]}e^{-\left((\rho\cos\theta)^2+(\rho\sin\theta)^2\right)}\left\vert\frac{d(x,y)}{d(\rho,\theta)}\right\vert d(\rho,\theta)\right)^{\frac{1}{2}}$$$$\longrightarrow\left(\iint_{\mathbb{R}^+\times[0,2\pi]}e^{-\left((\rho\cos\theta)^2+(\rho\sin\theta)^2\right)}\cdot\left\vert \begin{pmatrix} \dfrac{\partial}{\partial r}x(\rho,\theta) & \dfrac{\partial}{\partial\theta}x(\rho,\theta) \\ \dfrac{\partial}{\partial r}y(\rho,\theta) & \dfrac{\partial}{\partial\theta}y(\rho,\theta) \end{pmatrix} \right\vert d(\rho,\theta)\right)^\frac12$$
$$\mathbf{\text{My question}}$$
Would somebody please be able to explain the transformation to polar coordinates that happened? I'm not sure how that would be achieved.
$$\mathbf{\text{For clarification}}$$
$$\text{I usually talk like }\overset{\uparrow}{\text{as seen above}}\text{ because in my opinion,}$$$$\text{I feel like it is more needed if I need to clarify I step that I am taking,}$$$$\text{or if I need to recite a piece of info that has been previously stated}$$$$\text{in my question or in a previous question that I have asked,}$$$$\text{or if I am typing up a new header,}$$$$\text{although I can see why it can be annoying.}$$
$$\text{Sources}$$