Paper 3, Section II, B

(a) Given a space curve $\mathbf{r}(t)=(x(t), y(t), z(t)$, with $t$ a parameter (not necessarily arc-length), give mathematical expressions for the unit tangent, unit normal, and unit binormal vectors.

(b) Consider the closed curve given by

$x=2 \cos ^{3} t, \quad y=\sin ^{3} t, \quad z=\sqrt{3} \sin ^{3} t$

where $t \in[0,2 \pi)$.

Show that the unit tangent vector $\mathbf{T}$ may be written as

$\mathbf{T}=\pm \frac{1}{2}(-2 \cos t, \sin t, \sqrt{3} \sin t)$

with each sign associated with a certain range of $t$, which you should specify.

Calculate the unit normal and the unit binormal vectors, and hence deduce that the curve lies in a plane.

(c) A closed space curve $\mathcal{C}$ lies in a plane with unit normal $\mathbf{n}=(a, b, c)$. Use Stokes' theorem to prove that the planar area enclosed by $\mathcal{C}$ is the absolute value of the line integral

$\frac{1}{2} \int_{\mathcal{C}}(b z-c y) d x+(c x-a z) d y+(a y-b x) d z$

Hence show that the planar area enclosed by the curve given by $(*)$ is $(3 / 2) \pi$.

*Typos? Please submit corrections to this page on GitHub.*