Page 172 - Intro to Tensor Calculus

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a 11 0
I 27. For N =2 and a αβ = show that
0 a 22

R 1212 1 ∂ 1 ∂a 22 ∂ 1 ∂a 11
K = = − √ √ + √ .
a 2 a ∂u 1 a ∂u 1 ∂u 2 a ∂u 2

a 11 a 12
I 28. For N =2 and a αβ = show that
a 21 a 22

1 ∂ a 12 ∂a 11 1 ∂a 22 ∂ 2 ∂a 12 1 ∂a 11 a 12 ∂a 11
K = √ √ − √ + √ − √ − √ .
2 a ∂u 1 a 11 a ∂u 2 a ∂u 1 ∂u 2 a ∂u 1 a ∂u 2 a 11 a ∂u 1
Check your results by setting a 12 = a 21 = 0 and comparing this answer with that given in the problem 27.
I 29. Write out the Frenet-Serret formulas (1.5.112)(1.5.113) for surface curves in terms of Christoffel
symbols of the second kind.
I 30.
(a) Use the fact that for n =2 we have R 1212 = R 2121 = −R 2112 = −R 1221 together with e αβ ,e αβ the two
dimensional alternating tensors to show that the equation (1.5.110) can be written as

√ αβ 1 αβ
where αβ = and = √ e
R αβγδ = K αβ γδ ae αβ
a
are the corresponding epsilon tensors.
1 αβ γδ
(b) Show that from the result in part (a) we obtain R αβγδ = K.
4
Hint: See equations (1.3.82),(1.5.93) and (1.5.94).
I 31. Verify the result given by the equation (1.5.100).
2
I 32. Show that a αβ c αβ =4H − 2K.
I 33. Find equations for the principal curvatures associated with the surface

x = u, y = v, z = f(u, v).

I 34. Geodesics on a sphere Let (θ, φ) denote the surface coordinates of the sphere of radius ρ defined
by the parametric equations

x = ρ sin θ cos φ, y = ρ sin θ sin φ, z = ρ cos θ. (1)

Consider also a plane which passes through the origin with normal having the direction numbers (n 1 ,n 2 ,n 3 ).
This plane is represented by n 1 x+n 2 y+n 3z = 0 and intersects the sphere in a great circle which is described
by the relation
n 1 sin θ cos φ + n 2 sin θ sin φ + n 3 cos θ =0. (2)

This is an implicit relation between the surface coordinates θ, φ which describes the great circle lying on the
sphere. We can write this later equation in the form

−n 3
n 1 cos φ + n 2 sin φ = (3)
tan θ