Let $T(\theta): \Bbb R^3 \to \Bbb R^3$ be the rotation on the angle $\theta$ around $z$-axis. Write it in terms of coordinates $x, y, z$ and compute its Jacobian matrix.
I know the rotation matrix in terms of $\theta$, but how can I write it in terms of $x, y$ and $z$, is that $\cos \theta = x/y$?