Activity 13.7.4.
In this activity, we will work on calculating curl algebraically and interpreting it.
(a)
Consider the vector field \(\vF=\langle x,y,z\rangle\text{.}\) If we plot this vector field in any plane through the origin, we will see the vector field shown in Figure 13.7.19. This two-dimensional vector field has no rotation. The projection of \(\curl(\vF)(0,0,0)\) onto any direction therefore must give the zero vector. The only vector that has a zero projection in every direction is the zero vector. Verify this geometric argument for the curl of \(\vF\) by doing the calculations necessary to show that \(\curl(\langle{x,y,z}\rangle)=\vzero\text{.}\)
(b)
Now consider the vector field \(\vF=\langle{-y,x,0}\rangle\text{,}\) which is shown in Figure 13.7.20. The figure includes both an interactive three-dimensional plot of \(\vF\) as well as a two-dimensional plot of \(\vF\) in the yellow plane, which can be adjusted using the “Angle of Plane” slider.
(i)
Based on the figure, would you estimate the components of \(\curl(\vF)\) to be positive, negative, or zero at the origin? Does your answer change if you pick a different point?
(ii)
Calculate \(\curl(\vF)\) algebraically. Does the curl of this vector field vary depending on the point at which the curl is measured?
(iii)
Explain why the rotational strength of the vector field on a titled plane through the origin will be given by \(2\cos(\theta)\text{,}\) where \(\theta\) is the angle between \(\vk\) and the normal vector of the tilted plane.
(c)
In Figure 13.7.21 you can plot a vector field in a region around a point of your choosing in order to look at the rotational properties of the vector field. The check box in Figure 13.7.21 will show the curl vector at the base point specified so you can make sense of your vector field and its curl.
Use the figure to estimate the direction of \(\curl(\langle x-z,x^2+z,x+\sin(y)\rangle)\) at the point \((1,2,-1)\text{.}\) Confirm your estimate by calculating the curl at this point algebraically. Is there any point at which the direction of greatest rotational strength of this vector field has negative \(x\)-component? If there is, find such a point. If not, explain why not.
