Word Count : 366

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We are to check the local invertibility of $$ f(x,y) = (x^2 - y^2, \, 2xy) $$ at the point $(1, -1)$, and find a domain where $f$ is invertible. --- ### Step 1: Compute the Jacobian of $f(x, y)$ Let $$ u(x, y) = x^2 - y^2, \quad v(x, y) = 2xy $$ The Jacobian matrix $J_f(x, y)$ is $$ J_f(x, y) = \begin{bmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\[3pt] ____ ________ ________ ____ ______ __________ _____.
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