# Jacobian matrix for $f(x,y,z) := (4y, 3x^2-2\sin(yz), 2yz)$

I want to determine the Jacobian matrix $$J_f(x,y,z)$$ of the image $$f: \mathbb{R^3} \to \mathbb{R^3}$$ with $$f(x,y,z) := (4y, 3x^2-2\sin(yz), 2yz)$$.

So we have 3 coordinate functions $$f_1,f_2,f_3$$ with $$f_1(x,y,z) = 4y$$ $$f_2(x,y,z) = 3x^2 - 2\sin(yz)$$ $$f_3(x,y,z) = 2yz$$

So I get the following matrix for $$J_f(x,y,z)$$

$$\begin{pmatrix} 0 & 4 & 0\\ 6x & -2z\cos(zy) & -2\cos(yz)\\ 0 & 2z & 2y \end{pmatrix}$$

Is that correct or false?

And how can one find out the set of points $$a \in \mathbb{R^3}$$, for which the Jacobian Matrix $$J_f(a)$$ is not invertible?

Minor typo when you differentiate $$f_2$$ with respect to $$z$$,$$\begin{pmatrix} 0 & 4 & 0\\ 6x & -2z\cos(zy) & -2\color{blue}y\cos(yz)\\ 0 & 2z & 2y \end{pmatrix}$$
That is when $$4(6x)(2y)=0$$