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Curl - Multivariable Calculus

  1. Consider a 2D vector field F(x,y)=(P(x,y),Q(x,y),0)\mathbf{F}(x,y) = (P(x,y), Q(x,y), 0) where the curl is ∇×F=(0,0,∂Q∂x−∂P∂y)\nabla \times \mathbf{F} = (0, 0, \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}). If this field represents fluid flow and ∂Q∂x−∂P∂y=2ω\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 2\omega where ω>0\omega > 0, what does this indicate?
    Answer: The fluid has counterclockwise rotation with angular velocity ω\omega In 2D fluid flow, the zz-component of curl (∂Q∂x−∂P∂y\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) represents twice the angular velocity of local rotation. If this quantity is positive, it indicates counterclockwise rotation (using right-hand rule with positive zz-direction). Given that ∂Q∂x−∂P∂y=2ω\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 2\omega where ω>0\omega > 0, the curl magnitude is 2ω2\omega. Since curl magnitude equals twice the angular velocity, we have 2ω=2×angular velocity2\omega = 2 \times \text{angular velocity}, so the angular velocity is ω\omega. The positive sign indicates counterclockwise rotation. Choice A is wrong because positive curl indicates counterclockwise, not clockwise rotation. Choice C is wrong because it states angular velocity as 2ω2\omega instead of ω\omega. Choice D is wrong because non-zero curl definitely indicates net rotation.
  2. The curl of a vector field F\mathbf{F} at a point represents the axis and magnitude of rotation in the field. If ∇×F=(0,0,x2+y2)\nabla \times \mathbf{F} = (0, 0, x^2 + y^2) at all points, what can be concluded about the flow pattern?
    Answer: The field rotates about the zz-axis with angular velocity proportional to the square of distance from the zz-axis The curl vector (0,0,x2+y2)(0, 0, x^2 + y^2) points in the zz-direction with magnitude x2+y2=r2x^2 + y^2 = r^2, where rr is the distance from the zz-axis. The magnitude of curl equals twice the angular velocity, so ω=12(x2+y2)=12r2\omega = \frac{1}{2}(x^2 + y^2) = \frac{1}{2}r^2. Therefore, the angular velocity is proportional to the square of the distance from the zz-axis. The rotation increases with distance from the zz-axis, contradicting option D.

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