Proving dN/ds=−κT+τB: A Differential Geometry Homework Solution

[Murtuza Tipu]

Homework Statement

Currently revising for a differential geometry exam. The question I am working on is one of those types where the next part of the question follows from the last. I've gotten to the point where I have proven T⋅dNds=−κ,

Homework Equations

The next part is where I got stuck, which is to prove dN/ds=−κT+τB. I looked at the mark scheme

The Attempt at a Solution

it said "Follows from previous item, and B=T×N". I simply don't see how it follows, though.

 

[pasmith]

Homework Statement

Currently revising for a differential geometry exam. The question I am working on is one of those types where the next part of the question follows from the last. I've gotten to the point where I have proven T⋅dNds=−κ,

Homework Equations

The next part is where I got stuck, which is to prove dN/ds=−κT+τB. I looked at the mark scheme

The Attempt at a Solution

it said "Follows from previous item, and B=T×N". I simply don't see how it follows, though.

You know that $\frac{d\mathbf{N}}{ds}$ must be orthogonal to $\mathbf{N}$. Hence $\frac{d\mathbf{N}}{ds} = C\mathbf{T} + D\mathbf{B}$ for some $C(s)$ and $D(s)$. You have shown that $C = -\kappa$. How do you think you should go about finding $D$?