Can anyone explain to me the following concepts? Thank you very much.
1) a normal vector to the surface.
2) a principle unit normal vector to the surface.
Alright, if I am getting this right - Say you have a n-dimensional surface in $\displaystyle {\mathbb R}^m$ (certainly m>n). Choose a point on the surface and consider the tangent hyperplane at that point. This has n vectors as basis: $\displaystyle x_1,\ldots,x_n$. To complete this set to a basis of $\displaystyle {\mathbb R}^m$, you need m-n linearly independent vectors, orthogonal to that hyperplane: $\displaystyle \xi_1,\ldots,\xi_{m-n}$. These are called principal normal vectors. Divide by their lengths to get them to be unit: $\displaystyle \frac{1}{|\xi_1|}\xi_1,\ldots,\frac{1}{|\xi_{m-n}|}\xi_{m-n}$2) a principle unit normal vector to the surface.