The question is pretty much in the title; I'm about to teach my multivariable calculus students about orientations on surfaces, and I would like to be able to show them pictures. Any ideas?
Let's start with a parametrized surface. Any one will do, but I guess being orientable helps in this case.
Then calculate the unit normal, and then create a
To show the entire vector field for a surface on parametric form you can use a bunch of
As you can see, the arrows are equally spaced in the parameter space, which leads to uneven distribution of arrows on the surface; it might be worth transforming the parametrization into $f(x,y,z)=0$ form to get a nicer result.
The easiest thing to do is differentiate the field and use
Use a contour plot to visualize the region of a vector plot:
Plot a vector field over a particular region:
Create a contour plot of the vector plot's region:
Combine the vector and contour plots:
Just adding this answer for completion seeing as there is an out-of-the-box solution for this hidden in the documentation.
Essentially the following function:
(that can easily be tweaked to subsample or use arrows instead of lines) works in the previously mentioned examples:
and the Moebius band:
It isn't too hard to roll your own routine, of course:
It's not too hard to give the arrows depicting the normals some style: