# Paper 3, Section II, B

(a) Let $\mathbf{x}=\mathbf{r}(s)$ be a smooth curve parametrised by arc length $s$. Explain the meaning of the terms in the equation

$\frac{d \mathbf{t}}{d s}=\kappa \mathbf{n},$

where $\kappa(s)$ is the curvature of the curve.

Now let $\mathbf{b}=\mathbf{t} \times \mathbf{n}$. Show that there is a scalar $\tau(s)$ (the torsion) such that

$\frac{d \mathbf{b}}{d s}=-\tau \mathbf{n}$

and derive an expression involving $\kappa$ and $\tau$ for $\frac{d \mathbf{n}}{d s}$.

(b) Given a (nowhere zero) vector field $\mathbf{F}$, the field lines, or integral curves, of $\mathbf{F}$ are the curves parallel to $\mathbf{F}(\mathbf{x})$ at each point $\mathbf{x}$. Show that the curvature $\kappa$ of the field lines of $\mathbf{F}$ satisfies

$\frac{\mathbf{F} \times(\mathbf{F} \cdot \boldsymbol{\nabla}) \mathbf{F}}{F^{3}}=\pm \kappa \mathbf{b}$

where $F=|\mathbf{F}|$.

(c) Use $(*)$ to find an expression for the curvature at the point $(x, y, z)$ of the field lines of $\mathbf{F}(x, y, z)=(x, y,-z)$.