The Frenet-Settret formulae may be written d / d2 This is a system of first orde

Question

The Frenet-Settret formulae may be written d / d2 This is a system of first order linear ordinary differential equations. In general, a system of first order linear ordinary differential equations is of the form dx / ds (s) = M(s) x(s), where for each s 0, x(s) epsilon IRn is an n-component vector and M(s) is an n x n matrix. Let x0 epsilon IRn for each s 0, M(s) be an n times n matrix Assume the each matrix element of M(s) is continuous in s. It is true, but beyond the scope of this course to prove, that there is a unique solution to the initial value problem x(0) = xo, dx / ds (s) = M(s) x(s). In the case of the Frenet-Serret formulae, we also have that M(s) is antisymmetric, meaning that M(s)ij = -M(s)ji for all 1 i, j n and s 0 Assume that M(s) is antisymmetric. Prove that if x(s) obeys dx / ds (s) = M(s)x(s) for all s > 0, then |x(s)| is a constant, independent of s. Again assume that M(s) is antisymmetric. Prove that if both xa(s) and xb(s) solve the initial value problem dx / ds(s) = M(s) x(s), x(0) = x0, then xa(s) = xb(s) for all s 0. Let kappa(s) and tau(s) be continuous functions. Assume that T(s), N(s), B(s) obeys the Frenet-Serret formulae and that T(0), N(0) and B(0) are mutually perpendicular unit vectors. Prove that, for each s > 0, T(s), N(s) and B(s) are also mutually perpendicular unit vectors. Hint: You may use the "uniqueness of solutions to first order linear initial problems" result mentioned above.

Explanation / Answer

When one have a curve ß(s) which is parametrized by arc length (has natural parametrization) one is able to obtain the tangent, normal and binormal vectors by using Frenet-Serret frame equations: T=ß'(s), N=T'(s)|T'(s)|, B=T×N But are those formulas valid for non-regular parametrizations when one normalizes the tangent vector? T=ß'(s)|ß'(s)|, N and B are calculated as above.

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