5.8 The Curl of a Vector Field (continued)

Interpretation and Usage of Curl

Consider the vector field $\vF=\langle x,y,z\rangle$. If we plot this vector field in any plane through the origin, we will see the vector field shown in . 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 $\curl(\langle{x,y,z}\rangle)=\vzero$.

In this activity, we will work on calculating curl algebraically and interpreting it.

In 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 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 $(1,2,-1)$. 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.