Solution
= Solution
For $f=(f_1,\ldots,f_k)$, the <Jacobian matrix> is the matrix of its derivative in standard bases, with entries $(Df)_{ij}=\partial f_i/\partial x_j$. Here
$$
\boxed{Df(x,y)=\begin{pmatrix}e^x\sin y&e^x\cos y\\-e^{-y}\sin x&-e^{-y}\cos x\end{pmatrix}}.
$$