Paper 3, Section II, B

Vector Calculus | Part IA, 2017

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

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

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

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

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

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

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

F×(F)FF3=±κb\frac{\mathbf{F} \times(\mathbf{F} \cdot \boldsymbol{\nabla}) \mathbf{F}}{F^{3}}=\pm \kappa \mathbf{b}

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

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

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