Unconfusing Curl
By Jamie Ding
Updated 01/11/2007

Ever wonder what the meaning of it all really is? What is its purpose? Why does it even exist? Who made it in the first place! Or rather, was it made at all, or did it “evolve” from a more primitive form… Irregardlessly, Curl is confusing, so this manipulate module is meant to help you understand curl.

The vector field:

< xy, yz, y2 >

Is plotted on the left. At any given point, you can evaluate the curl of this vector field and it will result in a unique vector. The interesting thing is, this vector generated by the curl function on the vector field is also the axis of maximum torque. But what does this mean eh? This manipulate is meant to show you.

Hypothetically, if the vector field modeled, say, the flow of water in a pool, at any given point, the curl points in the direction of maximal rotational work. To better illustrate this, lets say the vector field in the manipulate module represents the flow of water at the base of the Niagara Falls. Suppose you want to generate electricity off this powerful natural wonder, and therefore want to mount generators in the water so that they will be spun as much as possible. Use the first 3 input fields to define the x, y, and z coordinates of the physical location in the water you wish to inspect. A red vector will appear, this is the curl vector, the axis of maximum torque. Now, using the last 3 input fields, define a new vector, this one will represent the direction you mount your generator’s shaft in. As you can see, as this new vector (the blue one) gets closer to the curl vector (the red one), the paddle, and therefore the generator shaft will spin faster.

Remember, the red one is the curl vector.

If you have any recommendations, please do submit them to the webmaster.

Notes:

1.         You can change the investigated vector field. At the top of the Mathematica file there are 3 definitions:

Fx[x_,y_,z_]=x y;
Fy[x_,y_,z_]=y z;
Fz[x_,y_,z_]=-y^2;

Change the functions on the right side of the equal sign to equal the x, y, and z components of your vector field function, using proper Mathematica notation of course. F(x,y,z)=<Fx, Fy, Fz>

2.         Mathematica 6.0 has issues with vector fields. You first have to load the graphics package (“<<Graphics`), and even then the first time you evaluate the “PlotVectorField3D” command, you will get errors, which disappear on the second evaluation.

3.         Also, when I say “% Maximum Torque” above the paddle wheel, it isn’t actually true. I simply use scalar vector projections to find out how “close” the direction of the blue vector points to the direction provided by the red vector. Therefore, I never actually figured out how to find the “spin” generated by the vector field, the “% Maximum Torque” bit is actually just an estimation of how close the directions of the vectors are. If anyone can tell me how to find the real torque generated by the vector field at a point in the direction of a vector, please email the webmaster. I’d love to know.