Page 130 - Intro to Tensor Calculus

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I 24. (parallel vector field) Imagine a vector field A = A (x ,x ,x ) which is a function of position.
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Assume that at all points along a curve x = x (t),i =1, 2, 3 the vector field points in the same direction,
~
we would then have a parallel vector field or homogeneous vector field. Assume A is a constant, then
~
k
∂ ~ A
dA = ∂x k dx =0. Show that for a parallel vector field the condition A i,k = 0 must be satisfied.
∂[ik, n] ∂ σ σ
I 25. Show that = g nσ +([nj, σ]+ [σj, n]) .
∂x j ∂x j ik ik
∂A r ∂A s
I 26. Show A r,s − A s,r = − .
∂x s ∂x r
I 27. In cylindrical coordinates you are given the contravariant vector components
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3
2
A = r A =cos θ A = z sin θ
(a) Find the physical components A r , A θ , and A z .

A rr A rθ A rz
(b) Denote the physical components of A i
,j , i, j =1, 2, 3, by A θr A θθ A θz
A zz .
A zr A zθ
Find these physical components.
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I 28. Find the covariant form of the contravariant tensor C = ijk A k,j .
k
Express your answer in terms of A .
,j
1
I 29. In Cartesian coordinates let x denote the magnitude of the position vector x i . Show that (a) x ,j = x j
x
1 1 2 1 −δ ij 3x i x j
(b) x ,ij = δ ij − x i x j (c) x ,ii = . (d) LetU = , x 6=0, and show that U ,ij = + and
x x 3 x x x 3 x 5
U ,ii =0.

I 30. Consider a two dimensional space with element of arc length squared

2 1 2 2 2 g 11 0
ds = g 11 (du ) + g 22 (du ) and metric g ij =
0 g 22
2
1
where u ,u are surface coordinates.
(a) Find formulas to calculate the Christoffel symbols of the first kind.
(b) Find formulas to calculate the Christoffel symbols of the second kind.

I 31. Find the metric tensor and Christoffel symbols of the first and second kind associated with the
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1
two dimensional space describing points on a cylinder of radius a. Let u = θ and u = z denote surface
coordinates where
x = a cos θ = a cosu 1
y = a sin θ = a sin u 1

z = z = u 2