Non-rotating vector fields with non-zero Curl

[ShayanJ]

In some texts the author tries to interpret operations like Curl.
Some say the curl of a vector field shows the amount of rotation of the vector field
But some of them say,if you put a wheel in a fluid velocity field which is like the vector field at hand,if it can rotate the wheel,then it has non-zero curl.
For example the field \vec{F}=z \hat{y} does not rotate by itself but if it shows a fluid velocity field,it can rotate wheels inside it and it has non-zero curl
But I'm just uneasy with the idea
maybe seeing more vector fields like this,ones which do not seem to rotate but have non-zero curl,make me better But I can't find more of such fields
Do you know one?
Thanks

 

[HallsofIvy]

The point is that the field itself is not "rotating" because all motion is in the x- direction. If this were a flow of water, every molecule would flow in a straight line. But the speed of the the water would vary and, if fact, all water molecules above the xy-plane would flow in the positive y direction, all molecules below would flow in the negative y diretction. That is, if we were to put a "water wheel" with axis along the x-axis, water flowing by above the xy-plane would push it in the positive y direction, water flowing by beneath would push it in the negatve z direction and so it would rotate clockwise- as seen looking at it form x< 0. That is an argument why curl f= \nabla f, which, here is
\left|\begin{array}{ccc}\vec{i} &amp; \vec{j} &amp; \vec{k} \\ \frac{\partial}{\partial x} &amp; \frac{\partial }{\partial y} &amp; \frac{\partial}{\partial z} \\ 0 &amp; z &amp; 0\end{array}\right|= -\vec{i}
is non zero even though every water molecule is moving in a straight line.

And, it works the other way. Imagine that you are holding a pan of water in front of you. Very slowly, so that you do not "slosh", move that pan around in a circle in front of you. Now, every water molecule in the pan is moving in a circle, relative to the ground but there is no "rotary motion". If you were to write out the equations for the motion of the water, in x, y, z terms, you would get curl 0.

 

[Studiot]

maybe seeing more vector fields like this,ones which do not seem to rotate but have non-zero curl,make me better But I can't find more of such fields
Do you know one?

Think about what you are asking.

The curl of any vector (a,b,c) is another vector (r,s,t)

for zero curl this requires r=s=t=0 since for any non zero r,s or t the modulus of the curl vector is √(r²+s²+t²)

However the velocity field in a fluid boundary layer is not (usaually) rotating, but has non zero curl. It may contain rotating vortices though.