Serret Frennet Formula |Part 3| Differential geometry and tensor Analysis |For B.Sc Students

Easiest derivation of Serret Frennet Formula For B.Sc Students
most important topic to examination point of view

How do you derive frenet Serret formula?


What is frenet Serret apparatus?


The tangent, normal, and binormal unit vectors, often called T, N, and B, or collectively the Frenet–Serret frame or TNB frame

T is the unit vector tangent to the curve, pointing in the direction of motion.

N is the normal unit vector, the derivative of T with respect to the arclength parameter of the curve, divided by its length.

B is the binormal unit vector, the cross product of T and N.

The Frenet–Serret formulas are:
{\displaystyle {\begin{aligned}{\frac {d\mathbf {T} }{ds}}&=\kappa \mathbf {N} ,\\{\frac {d\mathbf {N} }{ds}}&=-\kappa \mathbf {T} +\tau \mathbf {B} ,\\{\frac {d\mathbf {B} }{ds}}&=-\tau \mathbf {N} ,\end{aligned}}}
where d/ds is the derivative with respect to arclength, κ is the curvature, and τ is the torsion of the curve. The two scalars κ and τ effectively define the curvature and torsion of a space curve. The associated collection, T, N, B, κ, and τ, is called the Frenet–Serret apparatus