Let two vectors $z = [x\cdot t,y \cdot t,z \cdot t]$ , $b = [f(z),g(z),h(z),k(z)]$, where $f(z) =x,g(z)=y,h(z)=z,k(z)=t$. I try to calculate the jacobian $\frac{\partial{b}}{\partial{z}}$ without success. From the sizes of the vectors I suppose it should be $3 \times 4$ and also very symmetric, but I am not sure on what rule to use to approach for example $\frac{\partial{x}}{\partial{(xt)}}$ which would be the first element for example. Does my symbolism even have sense?
---------EDIT:--------------
Another approach that I followed: If the jacobian was $\frac{\partial z}{\partial b}$, then its computation would be trivial. My assumption is that intuitively $\frac{\partial z}{\partial b} * \frac{\partial b}{\partial z} = I$ where I is the identity matrix. However, following the same argument $\frac{\partial b}{\partial z} * \frac{\partial z}{\partial b} = I$ and there is no reason on why this should yield the same result. Any comments?