We need to show that velocity $\vec u$ of any point on rigid body and angular velocity of rigid body $\vec{\omega}$ are related as $\nabla \times \vec u = 2 \vec {\omega}$.
Initiate using $\vec u = \vec \omega \times \vec r$ where $\vec r$ is position of the point from axis of rotation, then I got $\vec u = \hat i (\omega_y z – \omega _z y) + \hat j (\omega_z x – \omega _x z) + \hat k (\omega _x y – \omega _y x)$
Now if only I consider $\vec \omega$ as constant and coordinates of point$(x, y, z)$ independant of each other then I get required result, but no such thing is mentioned in question to consider.
How to go about solving this without assuming these things. Also I think I used $x,y,z$ for point, then coordinate space is $(x', y', z')$ and so $\nabla = \hat i (\dfrac{\partial }{\partial x'}) + \hat j (\dfrac{\partial }{\partial y'}) +\hat k (\dfrac{\partial }{\partial z'})$
Best Answer
I am not sure what you mean by the coordinates being independent of eachother, but in any case we of course have $\partial_x x = 1,$ and $\partial_x y = \partial_x z = 0$, and analogously for the other partial derivatives simply directly from the definition.
On the other hamd, knowing that $$\vec u = \hat i (\omega_y z - \omega _z y) + \hat j (\omega_z x - \omega _x z) + \hat k (\omega _x y - \omega _y x),$$ and assuming that $\vec\omega = \vec\omega(x,y,z)$ depends on $x,y$ and $z$, we get
\begin{align} \nabla \times \vec u &= \begin{vmatrix} \hat i & \hat j & \hat z\\ \partial_x & \partial_y & \partial_z\\ \omega_y z - \omega _z y & \omega_z x - \omega _x z & \omega _x y - \omega _y x \end{vmatrix}\\[.2cm] &= \begin{pmatrix} \partial_y(\omega _x y - \omega _y x) - \partial_z(\omega_z x - \omega _x z)\\ \partial_z(\omega_y z - \omega _z y) - \partial_x(\omega _x y - \omega _y x)\\ \partial_x (\omega_z x - \omega _x z) - \partial_y(\omega_y z - \omega _z y) \end{pmatrix}\\[0.2cm] &= \begin{pmatrix} \big( (\partial_y\omega _x) y + \omega_x - (\partial_y\omega _y) x - 0\big) - \big((\partial_z\omega_z) x + 0 - (\partial_z\omega _x) z - \omega_x\big)\\ \big( (\partial_z\omega_y) z + \omega_y - (\partial_z\omega _z) y - 0 \big) - \big( (\partial_x\omega _x) y + 0 - (\partial_x\omega _y) x - \omega_y)\\ \big( (\partial_x\omega_z) x + \omega_z - (\partial_x\omega _x) z - 0 \big) - \big( (\partial_y\omega_y) z + 0 - (\partial_y\omega _z) y - \omega_z) \end{pmatrix}\\[0.2cm] &= 2\vec \omega + \begin{pmatrix} (\partial_y\omega_x)y-(\partial_y\omega_y)x-(\partial_z\omega_z)x+(\partial_z\omega_x)z\\ (\partial_z\omega_y)z-(\partial_z\omega_z)y-(\partial_x\omega_x)y + (\partial_x\omega_y)x\\ (\partial_x\omega_z)x - (\partial_x\omega_x)z -(\partial_y\omega_y)z + (\partial_y\omega_z)y\end{pmatrix}. \end{align} The latter vector is generally not equal to $0$ unless $\vec\omega$ is constant (take e.g. $\vec\omega(x,y,z) =(x,y,z)$ ), so the constancy condition was probably implicitly included in the question.