5.8 The Curl of a Vector Field

The Curl of a Vector Field

Introduction

Measuring the Circulation Density of Vector Field in

Measuring Rotation in Three Dimensions

Circulation Density in Three Dimensions

We would like to understand and measure rotation of a vector field near a particular point. In order to investigate this concept, we will look at some two-dimensional vector fields and think about whether the vector field shown will rotate a small pinwheel or spinner placed at a particular location. The sort of spinner we imagine is illustrated in . It consists of a central axis with a four-bladed paddle placed at one end of the axis. We imagine that the spinner is anchored at a point and the vector field, perhaps thought of as a fluid flow or wind velocity vector field, pushes against the blades of the spinner's paddle. In this activity, we will be trying to assess how the the spinner will rotate around the black axel (as an axis of rotation).

To begin our investigation of the rotation of a spinner in a vector field, we will look at the vector field $\vF$ in . As you think about these questions, draw an “X” at each of the points about which you are asked and consider the vector field as being the pattern of a wind blowing across the plane.

Draw an X at the origin to act as your spinner. Draw a vector on the top right blade of your spinner that represents how the wind will push on that blade. Next, draw a vector on each of the other blades of your spinner that represents how the wind will push on that blade.

Work through this exercise and explain your reasoning step by step.

$\langle -y,x\rangle$

$\langle x,y\rangle$

$\langle 1-x,xy\rangle$

Work through this exercise and explain your reasoning step by step.

Work through this exercise and explain your reasoning step by step.

Work through this exercise and explain your reasoning step by step.

Consider the vector field $\vF$ plotted in . 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 $(1,1,-2)$. 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 $(1,1,-2)$.

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 $z=-2$, as displayed in . 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 $z=-2$.

Next we will look at the vector field in the plane $y=1$, as displayed in . 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 $y=1$. Do you think the rotation in this figure is stronger or weaker than in ?

Finally, consider the trace of $\vF$ in the plane $x=1$, as displayed in . 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 $x=1$. How do you think the rotation in this figure compares (i.e., stronger or weaker) to that in and ?

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.

Compute the curl of $\vF=\langle x-y,y+2z,x^2\rangle$. Specifically, what is $\curl(\vF)$ at the point $(1,1,-2)$?

Compare the result of the curl calculation in part to your prediction from part. 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.