Paper 3, Section II, B

Vector Calculus | Part IA, 2021

(a) Given a space curve r(t)=(x(t),y(t),z(t)\mathbf{r}(t)=(x(t), y(t), z(t), with tt 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=2cos3t,y=sin3t,z=3sin3tx=2 \cos ^{3} t, \quad y=\sin ^{3} t, \quad z=\sqrt{3} \sin ^{3} t

where t[0,2π)t \in[0,2 \pi).

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

T=±12(2cost,sint,3sint)\mathbf{T}=\pm \frac{1}{2}(-2 \cos t, \sin t, \sqrt{3} \sin t)

with each sign associated with a certain range of tt, 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 C\mathcal{C} lies in a plane with unit normal n=(a,b,c)\mathbf{n}=(a, b, c). Use Stokes' theorem to prove that the planar area enclosed by C\mathcal{C} is the absolute value of the line integral

12C(bzcy)dx+(cxaz)dy+(aybx)dz\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)π(3 / 2) \pi.

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