I am working on a self study in an optimization textbook, and they are asking to "Determine the Jacobian of the following transformations:"
What is the best way to approach this problem? I think I'm confused, because I am used to seeing Jacobian questions with $x$ and $y$ with respect to $u$, $v$, and $w$.
$y_{1} = x_{1}$
$y_{2} = 5x_{3}$
$y_{3} = 4x_{2}^{2}-2x_{3}$
$y_{4} = x_{3}\sin(x_{1})$
On
The Jaccobian matrix is: $$\dfrac{\partial~(y_1,y_2,y_3,y_4)}{\partial~(x_1,x_2,x_3,x_4)}=\begin{bmatrix}\frac{\partial~y_{1}}{\partial~x_{1}} & \frac{\partial~y_{1}}{\partial~x_{2}}& \frac{\partial~y_{1}}{\partial~x_{3}}& \frac{\partial~y_{1}}{\partial~x_{4}} \\ \frac{\partial~y_{2}}{\partial~x_{1}} & \frac{\partial~y_{2}}{\partial~x_{2}}& \frac{\partial~y_{2}}{\partial~x_{3}}& \frac{\partial~y_{2}}{\partial~x_{4}}\\ \frac{\partial~y_{3}}{\partial~x_{1}} & \frac{\partial~y_{3}}{\partial~x_{2}}& \frac{\partial~y_{3}}{\partial~x_{3}}& \frac{\partial~y_{3}}{\partial~x_{4}}\\ \frac{\partial~y_{4}}{\partial~x_{1}} & \frac{\partial~y_{4}}{\partial~x_{2}}& \frac{\partial~y_{4}}{\partial~x_{3}}& \frac{\partial~y_{4}}{\partial~x_{4}}\end{bmatrix}$$
Yes, we treat the Jacobian matrix as the following matrix: $$ J_{ij}= \frac{\partial y_i}{\partial x_j} $$
So for example the first row of $J$ would be $\begin{bmatrix}1&0 &0 &0\end{bmatrix}$, since $y_1$ has derivative $1$ with respect to $x_1$, and $0$ with respect to the rest. You can find the other rows similarly.