A transformation T:(u,v)→(x,y)T: (u, v) \to (x, y)T:(u,v)→(x,y) is given by x=eucosv,y=eusinvx = e^u \cos v, y = e^u \sin vx=eucosv,y=eusinv. Find the Jacobian matrix J=∂(x,y)∂(u,v)J = \frac{\partial(x, y)}{\partial(u, v)}J=∂(u,v)∂(x,y) at (u,v)=(0,0)(u, v) = (0, 0)(u,v)=(0,0).
(1001)\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}(1001)
(0110)\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}(0110)
(111−1)\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}(111−1)
(e00e)\begin{pmatrix} e & 0 \\ 0 & e \end{pmatrix}(e00e)