Page 131 - Intro to Tensor Calculus

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I 32. Find the metric tensor and Christoffel symbols of the first and second kind associated with the
2
1
two dimensional space describing points on a sphere of radius a. Let u = θ and u = φ denote surface
coordinates where 1 2
x = a sin θ cos φ = a sin u cos u
1
y = a sin θ sin φ = a sin u sin u 2
z = a cos θ = a cos u 1

I 33. Find the metric tensor and Christoffel symbols of the first and second kind associated with the
two dimensional space describing points on a torus having the parameters a and b and surface coordinates
2
1
u = ξ, u = η. illustrated in the figure 1.3-19. The points on the surface of the torus are given in terms
of the surface coordinates by the equations
x =(a + b cosξ)cos η
y =(a + b cosξ)sin η
z = b sin ξ

r
i j k
i j m k
m j k i
i m k j
I 34. Prove that e ijk a b c u ,m + e ijk a b c u ,m + e ijk a b c u ,m = u e ijk a b c . Hint: See Exercise 1.3,
,r
problem 32 and Exercise 1.1, problem 21.
I 35. Calculate the second covariant derivative A i ,jk .

1 ∂ √ ij mn i
ij
I 36. Show that σ = √ gσ + σ
,j j
g ∂x mn
I 37. Find the contravariant, covariant and physical components of velocity and acceleration in (a) Cartesian
coordinates and (b) cylindrical coordinates.

I 38. Find the contravariant, covariant and physical components of velocity and acceleration in spherical
coordinates.

I 39. In spherical coordinates (ρ, θ, φ) show that the acceleration components can be represented in terms
of the velocity components as

2
v + v 2 φ v ρ v θ v 2 φ v ρ v φ v θ v φ
θ
f ρ =˙v ρ − , f θ =˙v θ + − , f φ =˙v φ + +
ρ ρ ρ tan θ ρ ρ tan θ
Hint: Calculate ˙v ρ , ˙v θ , ˙v φ .

i
i
I 40. The divergence of a vector A is A . That is, perform a contraction on the covariant derivative
,i
i
A i to obtain A . Calculate the divergence in (a) Cartesian coordinates (b) cylindrical coordinates and (c)
,j ,i
spherical coordinates.
I 41. If S is a scalar invariant of weight one and A i jk is a third order relative tensor of weight W,show
that S −W A i is an absolute tensor.
jk