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I need to compute the Jacobian for a transformation that maps parameters $p_1,...,p_n \to q_1,...,q_m$, $n\neq m$. For this, I need to compute the derivatives $\frac{\partial p_i}{\partial q_j}$. However, it would be much simpler to compute the derivatives $\frac{\partial q_i}{\partial p_j}$, i.e. the Jacobian of the inverse transformation, and then take the pseudo-inverse. I know this is possible when the Jacobian is invertible, but does it also work when it's not even a square matrix?

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If you take a surface $z=f(x,y)$ and re-write it as $x=g(z,y)$ then ${{\partial z} \over{ \partial x}}=f_x$ and ${{\partial x} \over{ \partial z}}=g_z$ will be reciprocal of each other since they are slopes (for a curve at fixed $y$) measured as $z$ changes with $x$, or as $x$ changes with $z$.

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