\[ \nabla\times\vec A=\begin{vmatrix}\hat a_x&\hat a_y&\hat a_z\\ \partial_x&\partial_y&\partial_z\\ A_x&A_y&A_z\end{vmatrix} \]
(1) \(A_x=2x^2+y^2\), \(A_y=xy-y^2\), \(A_z=0\)
- x-component: \(\partial_yA_z-\partial_zA_y=0-0=0\)
- y-component: \(\partial_zA_x-\partial_xA_z=0-0=0\)
- z-component: \(\partial_xA_y-\partial_yA_x=y-2y=-y\)
\(\nabla\times\vec A=-y\,\hat a_z\)
(2) \(A_x=yz\), \(A_y=4xy\), \(A_z=y\)
- x-component: \(\partial_y(y)-\partial_z(4xy)=1-0=1\)
- y-component: \(\partial_z(yz)-\partial_x(y)=y-0=y\)
- z-component: \(\partial_x(4xy)-\partial_y(yz)=4y-z\)
\(\nabla\times\vec A=\hat a_x+y\,\hat a_y+(4y-z)\,\hat a_z\)
