Activity 13.7.3. Estimating Curl in Three Dimensions.
(a)
Consider the vector field \(\vF\) plotted in Figure 13.7.14. You can adjust the size of the region around \((1,1,-2)\) over which the vector field is plotted using the “Zoom” slider. The “Density” slider allows you to adjust the number of vectors plotted. Try to identify any rotation in the three dimensional vector field plot at the point \((1,1,-2)\text{.}\) Write a sentence describing how a spinner placed at \((1,1,-2)\) would rotate, including along which axis it would rotate. Try to state your answer as a vector representing the rotational strength of the vector field at \((1,1,-2)\text{.}\)
(b)
You likely found it difficult to decide how you thought a spinner might rotate in this new, three-dimensional setting. Let’s look at the vector field in the plane \(z=-2\text{,}\) as displayed in Figure 13.7.15. Do you think a spinner placed on the red point would rotate clockwise, counterclockwise, or not rotate? If the spinner will rotate, you should think about what the axis of rotation would be and whether the rotation should be positive or negative. Summarize your result as a vector representing the rotational strength of \(\vF\) in the plane \(z=-2\text{.}\)
A vector field in two dimensions. The horizontal axis is labeled \(x\) and ranges from \(0\) to \(2\text{.}\) The vertical axis has the same range and is labeled \(y\text{.}\) There is a point marked at \((1,1)\text{.}\) Vectors point primarily down and are longer when closer to the horizontal axis. In the upper-left of the plot, the vectors point more down and left. In the lower-right of the plot, vectors point more down and right.
(c)
Next we will look at the vector field in the plane \(y=1\text{,}\) as displayed in Figure 13.7.16. Do you think a spinner placed on the red point would rotate clockwise, counterclockwise, or not rotate? If the spinner will rotate, you should think about what the axis of rotation would be and whether the rotation should be positive or negative. Summarize your result as a vector representing the rotational strength of your vector field in the plane \(y=1\text{.}\) Do you think the rotation in this figure is stronger or weaker than in Figure 13.7.15?
A vector field in two dimensions. The horizontal axis is labeled \(x\) and ranges from \(0\) to \(2\text{.}\) The vertical axis ranges from \(-3\) to \(-1\) and is labeled \(z\text{.}\) There is a point marked at \((1,-2)\text{.}\) Vectors in the right half of the figure are longer and point primarily up, although slightly to the left. Vectors in the left half of the figure are very short and point primarily right until around \(x=0.75\text{,}\) where they start to point up but are very short.
(d)
Finally, consider the trace of \(\vF\) in the plane \(x=1\text{,}\) as displayed in Figure 13.7.17. Do you think a spinner placed on the red point would rotate clockwise, counterclockwise, or not rotate? If the spinner will rotate, you should think about what the axis of rotation would be and whether the rotation should be positive or negative. Summarize your result as a vector representing the rotational strength of your vector field in the plane \(x=1\text{.}\) How do you think the rotation in this figure compares (i.e., stronger or weaker) to that in Figure 13.7.15 and Figure 13.7.15?
A vector field in two dimensions. The horizontal axis is labeled \(y\) and ranges from \(0\) to \(2\text{.}\) The vertical axis ranges from \(-3\) to \(-1\) and is labeled \(z\text{.}\) There is a point marked at \((1,-2)\text{.}\) Vectors on the left and bottom sides of the figure are slightly longer than vectors in the top and right portions of the figure. Vectors point predominantly up, although those in the lower-corner point up and left, with vectors becoming closer to upright as one moves to the upper-right corner.
(e)
Summarize your prediction to what you think the three-dimensional rotational strength of the vector field will be at the point \((1,1,-2)\) in the form of three-dimensional vector.
(f)
Compute the curl of \(\vF=\langle x-y,y+2z,x^2\rangle\text{.}\) Specifically, what is \(\curl(\vF)\) at the point \((1,1,-2)\text{?}\)
(g)
Compare the result of the curl calculation in part f to your prediction from part e. You likely found it difficult to estimate the magnitude, so your answer there may be incorrect. Hopefully, you did get the signs of the components and their relative strengths (i.e., which is biggest) correct. If you did not, go back and review the previous parts and explain why the calculated components match with the rotational strength for each of the three figures.
