How does the curl equation measure rotation?

[autodidude]

For a 2D vector field $${F}=P(x,y)\vec{i}+Q(x,y)\vec{j}$$

$$curl {F} = \frac{\partial Q}{\partial x}+\frac{\partial P}{\partial y}\vec{k}$$

So that's the rate of change of the j component of a field vector with respect to x plus the rate of change of the i component with respect to y...how does this measure the rotation about a point (x,y)?

 

[chiro]

Hey autodidude.

Is it possible for you to consider the cross product in terms A X B = ||a||*||b||*n*sin(theta) where n is a normal vector and theta is the angle between the two vectors?

 

[Studiot]

You should check your signs, there is a negative sign in the differential definition of the curl.

When you put the correct sign in you will see that the curl can be considered as the difference of flux entering/leaving along the x-axis and y axis.
ie it is the flux diverted from one axis to the other.
Rotation is the only way to achieve this.

You should note that the curl is twice the angular speed of rotation of a rigid body or twice the rotation rate of a vector field. In fluid mechanics it is called the vorticity or rot.

 

[Stephen Tashi]

I'm curious how people visualize the field itself rotating. I'm more familiar with the picture of of a small test apparatus consisting of "test" objects on the end of 3 perpendicular arms that turn on a "ball joint" where the arms meet. The field exerts a force on the objects to twist the apparatus about the ball joint. To find the curl exactly at the pivot, you take the limit of how the apparatus twists as the length of the arms approaches zero.

 

[psmt]

how does this measure the rotation about a point (x,y)?

I think what you're probably after is Stokes' theorem - try the Wikipedia page but skip to section 5, which they refer to as the Kelvin-Stokes theorem. By this, i mean that the integral ∫A.dr of the vector field around a closed loop intuitively has something to do with how the field is "rotating" on the surface enclosed. (Not so obvious if the enclosed surface chosen is not flat, but since the thread started with a 2D vector field i think we can restrict ourselves to flat surfaces).

As for motivational examples: it's easy to show (once you have the correct definition of curl!) that:
1) The curl of a constant vector field is zero (trivial, but confirms what you'd expect);
2) The curl of a radial field is zero - for example the curl of the vector field f(r)=r is zero, as you'd expect;
3) Consider a field which, if it corresponded to the velocity field of a fluid, would have the fluid rotating about the z-axis, for example A=(-y,x,0) in Cartesians. The curl of this field is constant and in the z-direction.
I think these examples at least show that the definition is sensible.

 

[autodidude]

Hey autodidude.

Is it possible for you to consider the cross product in terms A X B = ||a||*||b||*n*sin(theta) where n is a normal vector and theta is the angle between the two vectors?

I'm not sure how you would visualise the del operator as a vector...isn't the cross product just a handy formula just a handy way to calculate the curl though?

You should check your signs, there is a negative sign in the differential definition of the curl.

When you put the correct sign in you will see that the curl can be considered as the difference of flux entering/leaving along the x-axis and y axis.
ie it is the flux diverted from one axis to the other.
Rotation is the only way to achieve this.

You should note that the curl is twice the angular speed of rotation of a rigid body or twice the rotation rate of a vector field. In fluid mechanics it is called the vorticity or rot.

Sorry, my mistake! I did mean minus...(rate of change of the j component of a field vector with respect to x MINUS the rate of change of the i component of a field vector with respect to y). Could you maybe elaborate on the bolded bit? I'm still having a bit of trouble visualising this and relating it to the formula.

As for motivational examples: it's easy to show (once you have the correct definition of curl!) that:
1) The curl of a constant vector field is zero (trivial, but confirms what you'd expect);
2) The curl of a radial field is zero - for example the curl of the vector field f(r)=r is zero, as you'd expect;
3) Consider a field which, if it corresponded to the velocity field of a fluid, would have the fluid rotating about the z-axis, for example A=(-y,x,0) in Cartesians. The curl of this field is constant and in the z-direction.
I think these examples at least show that the definition is sensible.

Yeah, it does but at the moment I'm trying to get a picture just from what the equation itself says and try to visualise that