Consider $$\Phi: \mathbb{R}^2\to \mathbb{R}^3,~ \Phi(u,v)=\frac{1}{2}\begin{pmatrix}u+v\\u-v\\2uv\end{pmatrix}$$ I want to find the volume of the area $\Phi(\mathbb{R}^2)$ inside the cylinder $C:x^2+y^2<4$.
First I found the Jacobi-matrix of $\Phi$
$$J(u,v)_\Phi=\frac{1}{2}\begin{pmatrix}1&1\\ 1&-1\\ 2v&2u\end{pmatrix},~J(u,v)_\Phi^T=\begin{pmatrix}\frac{1}{2}&\frac{1}{2}& 1v\\ \frac{1}{2}&-\frac{1}{2}&1u\end{pmatrix}$$
So the Metric-Tensor and its determinant are
$$\mathscr{G}(\Phi)=J(u,v)_\Phi^T\cdot J(u,v)_\Phi=\begin{pmatrix}\frac{1+2v^2}{2}&vu\\ vu&\frac{1+2u^2}{2}\end{pmatrix},\quad \mathscr{g}(\Phi)=\frac{2v^2+2u^2+1}{4}$$
But Im not sure how to find the volume;