Mathematical Methods for Physics and Engineering : A Comprehensive Guide

10.11 EXERCISES

10.6 Prove that for a space curver=r(s), wheresis the arc length measured along
the curve from a fixed point, the triple scalar product
(
dr
ds

×

d^2 r

ds^2

)

·

d^3 r

ds^3

at any point on the curve has the valueκ^2 τ,whereκis the curvature andτthe
torsion at that point.
10.7 For the twisted space curvey^3 +27axz− 81 a^2 y= 0, given parametrically by

x=au(3−u^2 ),y=3au^2 ,z=au(3 +u^2 ),

show that the following hold:

(a) ds/du=3

√

2 a(1 +u^2 ), wheresis the distance along the curve measured from

the origin;

(b) the length of the curve from the origin to the Cartesian point (2a, 3 a, 4 a)is

4

√

2 a;

(c) the radius of curvature at the point with parameteruis 3a(1 +u^2 )^2 ;

(d) the torsionτand curvatureκat a general point are equal;

(e) any of the Frenet–Serret formulae that you have not already used directly

are satisfied.

10.8 The shape of the curving slip road joining two motorways, that cross at right
angles and are at vertical heightsz=0andz=h, can be approximated by the
space curve

r=

√

2 h

π

ln cos

(zπ

2 h

)

i+

√

2 h

π

ln sin

(zπ

2 h

)

j+zk.

Show that the radius of curvatureρof the slip road is (2h/π)cosec (zπ/h)at
heightzand that the torsionτ=− 1 /ρ. To shorten the algebra, setz=2hθ/π
and useθas the parameter.
10.9 In a magnetic field, field lines are curves to which the magnetic inductionBis
everywhere tangential. By evaluatingdB/ds,wheresis the distance measured
along a field line, prove that the radius of curvature at any point on a line is
given by

ρ=

B^3

|B×(B·∇)B|

.

10.10 Find the areas of the given surfaces using parametric coordinates.

(a) Using the parameterisationx=ucosφ,y=usinφ,z=ucot Ω, find the

sloping surface area of a right circular cone of semi-angle Ω whose base has

radiusa.Verifythatitisequalto^12 ×perimeter of the base×slope height.

(b) Using the same parameterization as in (a) forxandy, and an appropriate

choice forz, find the surface area between the planesz=0andz=Zof

the paraboloid of revolutionz=α(x^2 +y^2 ).

10.11 Parameterising the hyperboloid

x^2

a^2

+

y^2

b^2

−

z^2

c^2

=1

byx=acosθsecφ,y=bsinθsecφ,z=ctanφ, show that an area element on

its surface is

dS=sec^2 φ

[

c^2 sec^2 φ

(

b^2 cos^2 θ+a^2 sin^2 θ

)

+a^2 b^2 tan^2 φ

] 1 / 2

dθ dφ.