vorticity not zero - why

I am quite new to fluid dynamics and I cannot seem to understand the concept of vorticity. It is defined as $\nabla\times\vec{v}$ where $\vec{v}$ is the velocity vector. Now (working in cartesian coordinates) velocity: $\vec{v}=\frac{{\rm d} \vec{r}}{{\rm d} t}$ so that $\nabla\times\vec{v}=\nabla\times\frac{{\rm d} \vec{r}}{{\rm d} t}$. And (assuming interchangeable derivatives as is usual) $\nabla\times\vec{v}=\frac{{\rm d}}{{\rm d} t}\nabla\times\vec{r}\equiv 0$.

So because vorticity is even considered as a quantity and thus there have to be cases when it is non-zero, I have made an error. I can think of two sources. One, the spatial curl derivative in $\nabla$ is with respect to different variables (I doubt that). Two, the derivative is not interchangeable. Or is there another reason? Please try to explain in terms of definition of velocity as a time derivative of a position vector, rather than as it is usually done in terms of a given velocity field $(v_x(t,\vec{r}),v_y(t,\vec{r}),v_z(t,\vec{r}))$ without any apparent relation to $\vec{r}$. Thank you.