## Calculus (3rd Edition)

$1$.
To find the Jacobian, we calculate the determinant of the 2x2 matrix $\frac{\partial(x, y)}{\partial(r, t)}$ as follows $$\operatorname{Jac}(G)=\frac{\partial(x, y)}{\partial(r, t)}=\left|\begin{array}{ll} {\frac{\partial x}{\partial r}} & {\frac{\partial x}{\partial t}} \\ {\frac{\partial y}{\partial r}} & {\frac{\partial y}{\partial t}} \end{array}\right|=\left|\begin{array}{ll} { \sin t} & {r\cos t} \\ {1} & {\sin t} \end{array}\right| =\sin^2 t-r\cos t.$$ Now, at the point $(r,t)= (1,\pi)$, we have $\operatorname{Jac}(G)=\sin^2 \pi-\cos \pi=1.$