I am stuck with the following problem that says :
If$$\begin{align} u^3+v+w&=x+y^2+z^2,\\u+v^3+w&=x^2+y+z^2,\\u+v+w^3&=x^2+y^2+z, \end{align}$$then prove that $$\frac{\partial(u,v,w)}{\partial(x,y,z)}=\frac{1-4(xy+yz+zx)+16xyz}{2-3(u^2+v^2+w^2)+27u^2v^2w^2}$$
My try: I have to compute the value of $$\begin{vmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} &\frac{\partial u}{\partial z} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} &\frac{\partial v}{\partial z} \\ \frac{\partial w}{\partial x} & \frac{\partial w}{\partial y} &\frac{\partial w}{\partial z} \end{vmatrix}$$.
For that I need to convert $u,v,w$ in terms of $x,y,z$. I am unable to do that as things are getting complicated.
From the given set of equations we get, $$3u^2 \frac{\partial u}{\partial x}+ \frac{\partial v}{\partial x}+ \frac{\partial w}{\partial x}=1 \\ \frac{\partial u}{\partial x}+ 3v^2\frac{\partial v}{\partial x}+ \frac{\partial w}{\partial x}=2x \\ \frac{\partial u}{\partial x}+ \frac{\partial v}{\partial x}+ 3w^2\frac{\partial w}{\partial x}=2x \implies\\ \begin{pmatrix} 3u^2 & 1 &1\\ 1& 3v^2& 1\\ 1& 1 & 3w^2 \end{pmatrix} \begin{pmatrix} \frac{\partial u}{\partial x}\\ \frac{\partial v}{\partial x}\\ \frac{\partial w}{\partial x} \end{pmatrix}=\begin{pmatrix} 1\\ 2x\\ 2x \end{pmatrix}$$
Can someone give me a detailed explanation? Thanks in advance for your time.
You've already expressed the partial derivatives with respect to $x$ as a matrix-vector equation: $$\begin{pmatrix} 3u^2 & 1 &1\\ 1& 3v^2& 1\\ 1& 1 & 3w^2 \end{pmatrix} \begin{pmatrix} \frac{\partial u}{\partial x}\\ \frac{\partial v}{\partial x}\\ \frac{\partial w}{\partial x} \end{pmatrix}=\begin{pmatrix} 1\\ 2x\\ 2x \end{pmatrix}.$$ Doing the same for partial derivatives with respect to $y$ and $z$, we get $$\begin{pmatrix} 3u^2 & 1 &1\\ 1& 3v^2& 1\\ 1& 1 & 3w^2 \end{pmatrix} \begin{pmatrix} \frac{\partial u}{\partial y}\\ \frac{\partial v}{\partial y}\\ \frac{\partial w}{\partial y} \end{pmatrix}=\begin{pmatrix} 2y\\ 1\\ 2y \end{pmatrix}$$
and
$$\begin{pmatrix} 3u^2 & 1 &1\\ 1& 3v^2& 1\\ 1& 1 & 3w^2 \end{pmatrix} \begin{pmatrix} \frac{\partial u}{\partial z}\\ \frac{\partial v}{\partial z}\\ \frac{\partial w}{\partial z} \end{pmatrix}=\begin{pmatrix} 2z\\ 2z\\ 1 \end{pmatrix}.$$ The key step remaining is to combine these three into a single matrix-matrix equation: $$\begin{pmatrix} 3u^2 & 1 &1\\ 1& 3v^2& 1\\ 1& 1 & 3w^2 \end{pmatrix} \begin{pmatrix} \frac{\partial u}{\partial x}&\frac{\partial u}{\partial y}&\frac{\partial u}{\partial z}\\ \frac{\partial v}{\partial x}&\frac{\partial v}{\partial y}&\frac{\partial v}{\partial z}\\ \frac{\partial w}{\partial x}&\frac{\partial w}{\partial y}&\frac{\partial w}{\partial z} \end{pmatrix}=\begin{pmatrix} 1&2y&2z\\ 2x&1&2z\\ 2x&2y&1 \end{pmatrix}.$$ Taking determinants, we get $$(2-3(u^2+v^2+w^2)+27u^2v^2w^2)\left\vert\frac{\partial(u,v,w)}{\partial(x,y,z)}\right\vert=1-4(xy+yz+zx)+16xyz$$ and the result follows immediately.