Page 174 - Intro to Tensor Calculus

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I 35. Find the differential equations defining the geodesics on the surface of a cylinder.

I 36. Find the differential equations defining the geodesics on the surface of a torus. (See problem 13,
Exercise 1.3)
I 37. Find the differential equations defining the geodesics on the surface of revolution

x = r cos φ, y = r sin φ, z = f(r).

Note the curve z = f(x) gives a profile of the surface. The curves r = Constant are the parallels, while the
curves φ = Constant are the meridians of the surface and

02
2
2
2
2
ds =(1 + f ) dr + r dφ .
I 38. Find the unit normal and tangent plane to an arbitrary point on the right circular cone
x = u sin α cos φ, y = u sin α sin φ, z = u cos α.

This is a surface of revolution with r = u sin α and f(r)= r cot α with α constant.
I 39. Let s denote arc length and assume the position vector ~(s) is analytic about a point s 0 . Show that
r
h 2 h 3
r
r
~ (s 0 )+ ··· about the point s 0 ,with h = s − s 0 is
r
0
00
the Taylor series ~(s)= ~(s 0 )+ h~r (s 0 )+ ~ (s 0 )+ r 000
2! 3!
2 ~
1
~
1 3
0 ~
2 ~
~
r
r
given by ~(s)= ~(s 0 )+ hT + κh N + h (−κ T + κ N + κτB)+ ··· which is obtained by differentiating
2 6
the Frenet formulas.
I 40.
(a) Show that the circular helix defined by x = a cos t, y = a sin t, z = bt with a, b constants, has the
property that any tangent to the curve makes a constant angle with the line defining the z-axis.
~
(i.e. T · b e 3 =cos α = constant.)
~
(b) Show also that N · b e 3 = 0 and consequently b e 3 is parallel to the rectifying plane, which implies that
~
~
b e 3 = T cos α + B sin α.
(c) Differentiate the result in part (b) and show that κ/τ =tan α is a constant.
I 41. Consider a space curve x i = x i (s) in Cartesian coordinates.
~ p
dT 0 0
i i
(a) Show that κ = = x x
ds
1
0
r
r
0 00 000
(b) Show that τ = e ijk x x x . Hint: Consider ~ · ~ × ~ 000
r
00
i j
k
κ 2
I 42.
(a) Find the direction cosines of a normal to a surface z = f(x, y).
(b) Find the direction cosines of a normal to a surface F(x, y, z)= 0.
(c) Find the direction cosines of a normal to a surface x = x(u, v),y = y(u, v),z = z(u, v).
I 43. Show that for a smooth surface z = f(x, y) the Gaussian curvature at a point on the surface is given
by
2
f xx f yy − f xy
K = .
2
2
(f + f +1) 2
x
y