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?
The easiest thing to do is differentiate the field and use VectorPlot3D and ContourPlot3D to show orthogonality of these. This is from Documentation Center. I will change this a bit from original to polish graphics for your lecture. These are not unit vectors though - do u really need unit ones? It can be done too if you need.
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:
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.
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: