Mathematical Methods for Physics and Engineering - Matematica.NET

10.11 EXERCISES10.6 Prove that for a space curve r = r(s), where s is the arc length measured alongthe curve from a fixed point, the triple scalar product( ) drds × d2 r· d3 rds 2 ds 3at any point on the curve has the value κ 2 τ,whereκ is the curvature and τ thetorsion at that point.10.7 For the twisted space curve y 3 +27axz − 81a 2 y = 0, given parametrically byx = au(3 − u 2 ), y =3au 2 , z = au(3 + u 2 ),show that the following hold:(a) ds/du =3 √ 2a(1 + u 2 ), where s is the distance along the curve measured fromthe origin;(b) the length of the curve from the origin to the Cartesian point (2a, 3a, 4a) is4 √ 2a;(c) the radius of curvature at the point with parameter u is 3a(1 + u 2 ) 2 ;(d) the torsion τ and curvature κ at a general point are equal;(e) any of the Frenet–Serret formulae that you have not already used directlyare satisfied.10.8 The shape of the curving slip road joining two motorways, that cross at rightangles and are at vertical heights z =0andz = h, can be approximated by thespace curve√2hr =π( zπ)√2hln cos i +2h π( zπ)ln sin j + zk.2hShow that the radius of curvature ρ of the slip road is (2h/π)cosec (zπ/h) atheight z and that the torsion τ = −1/ρ. To shorten the algebra, set z =2hθ/πand use θ as the parameter.10.9 In a magnetic field, field lines are curves to which the magnetic induction B iseverywhere tangential. By evaluating dB/ds, wheres is the distance measuredalong a field line, prove that the radius of curvature at any point on a line isgiven byB 3ρ =|B × (B · ∇)B| .10.10 Find the areas of the given surfaces using parametric coordinates.(a) Using the parameterisation x = u cos φ, y = u sin φ, z = u cot Ω, find thesloping surface area of a right circular cone of semi-angle Ω whose base hasradius a. Verifythatitisequalto 1 ×perimeter of the base ×slope height.2(b) Using the same parameterization as in (a) for x and y, and an appropriatechoice for z, find the surface area between the planes z =0andz = Z ofthe paraboloid of revolution z = α(x 2 + y 2 ).10.11 Parameterising the hyperboloidx 2a + y22 b − z22 c =1 2by x = a cos θ sec φ, y = b sin θ sec φ, z = c tan φ, show that an area element onits surface isdS =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 ] 1/2dθ dφ.371