Exercises
7-1. Sketch the given surfaces (i)
y^2
a^2 +z^2
b^2 =x^2
c^2 (ii)z^2
a^2 +x^2
b^2 =y^2c^2 , a > b > c
7-2. Sketch the given surfaces (i)
y^2
a^2 +z^2
b^2 =x
c (ii)z^2
a^2 +x^2
b^2 =yc,a > b > c
7-3. Sketch the given surfaces defined by the parametric equations
(i) x−x 0 =u, y −y 0 =v, z −z 0 =c(
u^2
a^2 +v^2
b^2)(ii) x−x 0 =u, y −y 0 =v, z −z 0 =c(u 2
a^2 −v^2
b^2)7-4. The curve r =r (t) = αcos ωt ˆe 1 +αsin ωt eˆ 2 +βteˆ 3 , where α, β and ωare constants,
describes a circular helix of radius α. For this space curve calculate the following
quantities.
(a) The unit tangent vector ˆet
(b) The unit normal vector ˆen
(c) The unit binormal vector ˆeb
(d) The curvature κ
(e) The torsion V
7-5. If r =r (t)denotes a space curve, show that the curvature is given by
κ=√
(r ′·r ′)(r ′′·r ′′)−(r ′·r ′′)^2
(r ′·r ′)^3 /^2where ′=dtd denotes differentiation with respect to the argument of the function.
7-6. If r =r (x) = xˆe 1 +y(x)ˆe 2 is the position vector describing a curve in the
x, y −plane, show that the curvature is given by
κ= |y′′|
(1 + (y′)^2 )^3 /^2where ′=dxd denotes differentiation with respect to the argument of the function.
7-7.
(a) For r =r (s)the position vector of a curve, show that drds ×d
(^2) r
ds^2 =κeˆb.
(b) For r =r (s)the position vector of a curve, show that
∣∣
∣∣dr
ds ×d^2 r
ds^2∣∣
∣∣=κ.