Thread: Real Analyis, vector valued function,differentiability?

please help with detailed steps.
thank a lot~

Hey cyhui.

This reminds of frenet frame considering of a normal vector, a bi-normal and a tangent vector and these things are discussed in curved geometries. Take a look at this:

Frenet

You'll notice that there are expressions for Tangent, Normal, and Bi-Normal vectors and all you have to show is that the Tangent vector is non-zero. You have a non-zero normal vector which means that the Bi-Normal and Tangent Vectors are also non-zero.

Sorry, i get confused about it. I haven't learn this before.
Still don't understand how to find the function out.

The T in wikipedia relates to <x'(0),y'(0),x'(0)> and the normal vector is non-zero (this is the N) which in this case is (a,b,c) and from the Wiki-page we know that T = F'(x)/||F'(x)|| (which is what the grad refers to).

Since the normal vector N which is computed from N = T X B is non-zero, the only way this can happen is if B and T are non-zero and also not linearly dependent on each other. So this means that your gradF is non-zero for this reason.