Potential function in fluids

If the flow is rotational: $$nabla times F = frac{partial F_x}{partial y} - frac{partial F_y}{partial x} neq 0$$

But if $F = nabla phi$, where $phi$ is a potential scalar function,

$$frac{partial F_x}{partial y} = frac{partial^2 phi}{partial x partial y}$$

$$frac{partial F_y}{partial x} = frac{partial^2 phi}{partial y partial x}$$

As the order of the second derivative doesn't matter, the curl should be zero.

So, if the flow is rotational, it can not be expressed as a gradient of a potential function. While I can always write:

$$dF = F_xdx + F_ydy$$

$F_x$ and $F_y$ are not the components of the gradient of any scalar function. For an exact differential by definition, it should be possible to write:

$$dphi = frac{partial phi}{partial x}dx + frac{partial phi}{partial y}dy$$

for some scalar function $phi$