Given the following vector, \begin{align} F(x(t),y(t),z(t)) &= \begin{bmatrix} \omega_1^2 x_o\cos(\omega_1 t) \\ \omega_2y_0\sin(\omega_2 t)\\ 0\\ \end{bmatrix} \end{align}
I want to determine the curl: $\nabla \times F. $ Even though it should be easy, I could not come along. The first problem is, I do not get if coordinates used can be considered as polar or cylindrical. I think they can't be either, since the angle function is different for $\cos$ and $\sin$. But it is difficult to consider them as cartesian coordinates as well, so that I do not see how to calculate the curl. I want to prove or reject that $F$ is conservative. Instead of finding a potential and prove that the gradient of that function is equal to the given force, I wanted to rather try to find the curl of the function $F.$ Can somebody provide some insight or a solution proposal? Thanks.
