Multivariable Calculus Quiz: Curl
2 questions · exam conditions
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CurlQuestion 1 of 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?

The field has no rotational component anywhere in the domain
The field rotates about the zz-axis with angular velocity proportional to distance from the zz-axis
The field rotates about the zz-axis with angular velocity proportional to the square of distance from the zz-axis
The field has maximum rotation at the origin and decreases with distance
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Multivariable Calculus Quiz

Multivariable Calculus Quiz: Curl

Practice Curl in Multivariable Calculus with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Curl, giving you a quick way to practice the rules, question types, and explanations that matter most for Multivariable Calculus.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

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?

  1. The field has no rotational component anywhere in the domain
  2. The field rotates about the zz-axis with angular velocity proportional to distance from the zz-axis
  3. The field rotates about the zz-axis with angular velocity proportional to the square of distance from the zz-axis (correct answer)
  4. The field has maximum rotation at the origin and decreases with distance
Explanation: 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.

Question 2

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?

  1. The fluid has clockwise rotation with angular velocity ω\omega
  2. The fluid has counterclockwise rotation with angular velocity ω\omega (correct answer)
  3. The fluid has counterclockwise rotation with angular velocity 2ω2\omega
  4. The fluid has no net rotation but non-zero shear
Explanation: 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.