Jacobian Matrix

Consider the function \(f: \mathbb{R}^2\to \mathbb{R}^2\) defined by \(\cv{x\\y}\mapsto \cv{x+3\,y \\ x^2-y^2+1}\).

Compute the Jacobian matrix \(J(x,y)\) and determine its value in the point \((x,y)=(0,-3)\).

\(J(x,y) ={}\)

\(\matrix{\Box & \Box \\ \Box & \Box}\)

\(J(x,y)\Biggl|_{(0,-3)} ={}\)

\(\matrix{\Box & \Box \\ \Box & \Box}\)