Are unidirectional vector fields conservative?

Question

There is no rotation for unidirectional (all the vectors pointing in the same direction whatever be their magnitudes) vector fields. Hence their curl must be zero. Hence all unidirectional vector fields MUST be conservative. Am I correct?

Answer 1

In addition to the computational answer of 'Lord Shark the Unknown', you can also look at this graphical example from Wikipedia:

Suppose we now consider a slightly more complicated vector field:

$${\mathbf{F}}(x,y,z) = - x^2 \hat{\mathbf{y}}$$

Its plot:

We might not see any rotation initially, but if we closely look at the right, we see a larger field at, say, $x = 4$ than at $x = 3$. Intuitively, if we placed a small paddle wheel there, the larger "current" on its right side would cause the paddlewheel to rotate clockwise, which corresponds to a curl in the negative $z$ direction. By contrast, if we look at a point on the left and placed a small paddle wheel there, the larger "current" on its left side would cause the paddlewheel to rotate counterclockwise, which corresponds to a curl in the positive $z$ direction. [...]

Answer 2

So one has a vector field, which in Cartesian coordinates is $$(f(x,y,z),0,0).$$ So its curl is $$\left(0,\frac{\partial f}{\partial z},-\frac{\partial f}{\partial y}\right).$$ Looks likely to be nonzero to me.