I am pretty sure that the problem I am working on is supposed to be solved by applying the multivariate chain rule, but despite asking my TA, consulting the text and my notes, searching for quite a while online and working on the problem for a few hours, I've made pretty much no progress. As in, I still hardly know how to begin working the problem. I get the feeling it isn't even that hard to compute, but I just don't understand the technique.

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Here is the problem:

Let $\begin{pmatrix}y_1 \\ y_2\end{pmatrix} = \vec{y} = \vec{f}(\vec{x}) = \begin{pmatrix}x_1^2 - x_2^2 \\ 2x_1x_2\end{pmatrix}$, where $\vec{x} = \begin{pmatrix}x_1 \\ x_2\end{pmatrix}$.

Define $\vec{x}$ as a function of $\vec{y}$ implicitly by $\vec{y}$ = $\vec{f}$($\vec{x}$). That is, let $\vec{x}$ = $\vec{g}$($\vec{y}$) where $\vec{y}$ = $\vec{f}$($\vec{g}$($\vec{y}$)).

Derive equations that $\frac{∂x_1}{∂y_1}$ and $\frac{∂x_2}{∂y_1}$ must satisfy, and then solve them for $\frac{∂x_1}{∂y_1}$ and $\frac{∂x_2}{∂y_1}$. Do the same for $\frac{∂x_1}{∂y_2}$ and $\frac{∂x_2}{∂y_2}$.

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This is the part of the problem that I am stuck on. There are a few subsequent parts as well, but I am pretty confident I can figure them out if I could just get this part down. My TA has given me some (very, very) vague hints, and they didn't help much at all.

I don't like to ask this typically, but if someone could run through the calculation for one of the equations the question refers to, or even do the calculation on a similar but different problem, I think that would be the best way for me to understand what is going on. I say this because I already know the chain rule and everything, the only thing that has stumped me is actually applying it.

If you would prefer to give a hint or something else, that is fine too. Any help is appreciated!

Edit: I still would really like some help with this. I've asked my professor and my TA and posted asking for help here and on Reddit, and so far I haven't gotten a really concrete hint so to speak from any of those sources.

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    $\begingroup$ Hint: ditch the matrix notation, and write out the two equations, so $y_1=x_1^2-x_2^2$ and $y_2=2x_1x_2$. I think if you differentiate both w.r.t $X_1$ and $X_2$ it will go better... $\endgroup$ – eSurfsnake Feb 7 at 3:47
  • $\begingroup$ I actually had to differentiate y1 w.r.t x1 and x2 and so on for a previous part of the question. How can I use those derivatives to find equations that dx1/dy1 and all the rest must satisfy though? $\endgroup$ – KRob Feb 7 at 3:51
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    $\begingroup$ Do you know about Jacobian matrices? $\endgroup$ – amd Feb 7 at 4:40
  • $\begingroup$ Yes, I know how to write them if I have the partial derivatives. Is there some way to convert the Jacobian matrix for $\vec{y}_{\vec{x}}$ to the Jacobian matrix for $\vec{x}_{\vec{y}}$? $\endgroup$ – KRob Feb 7 at 5:19
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    $\begingroup$ The function $g$ is the inverse function of $f$, so the Jacobian matrix of $g$ is the inverse of the Jacobian matrix of $f$. (Which one sees by applying the chain rule to the equation $f(g(y))=y$.) $\endgroup$ – Hans Lundmark Feb 7 at 8:44

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