In the Frenet Formula, I can see why the derivative of the tangent vector is a function of the curvature times the normal vector, and the derivative of the binormal vector is a negative function of the torsion times the normal vector, (where B=TxN). But I don't quite understand why the derivative of the normal vector is function of BOTH the curvature and torsion.
Apparently, there is a "matrix symmetry" proof of this theorem using these vectors and their derivatives. But I'm looking for a calculus-based proof (if it exists). Can you express the normal as some product of the binormal and tangent vectors and then use a product rule? Or can you do some kind of convolution, inversion-type manipulation.