I'm doing a calculus in which I have a vector $$\displaystyle V=\begin{pmatrix} \displaystyle a\left(x+y\right)\\ \displaystyle -ay\\ \displaystyle w_0 \end{pmatrix}$$
The thing is I need to compute the jacobian of $V$ which I found was $$ J=\begin{pmatrix} \displaystyle a &a&0\\ \displaystyle 0 & -a & 0\\ \displaystyle 0 & 0 & 0 \end{pmatrix} $$ and also $$\displaystyle \vec{\text{grad}}\left(\vec{V}^2\right)=\begin{pmatrix} \displaystyle 2a^2x+2a^2y\\ \displaystyle 2a^2y\\ \displaystyle 0 \end{pmatrix}$$ and $$\displaystyle \vec{\text{rot}}\left(\vec{V}\right)\wedge \vec{V}=\begin{pmatrix} \displaystyle 0\\ \displaystyle 0\\ \displaystyle -a \end{pmatrix}\wedge\begin{pmatrix} \displaystyle a\left(x+y\right)\\ \displaystyle -ay\\ \displaystyle w_0 \end{pmatrix}=\begin{pmatrix} \displaystyle -a^2 y\\ \displaystyle -a^2\left(x+y\right)\\ \displaystyle 0 \end{pmatrix}$$ where $\wedge$ is the cross product.
I know that I should obtain $$J.V=\vec{\text{rot}}\left(\vec{V}\right)\wedge \vec{V}+\vec{\text{grad}}\left(\frac{\vec{V}^2}{2}\right)$$ but here I obtain $$ \begin{pmatrix} \displaystyle a^2x\\ \displaystyle -a^2 x\\ \displaystyle 0 \end{pmatrix}$$
so where do i made a mistake please ?