Page 128 - Intro to Tensor Calculus

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EXERCISE 1.4

I 1. Find the nonzero Christoffel symbols of the first and second kind in cylindrical coordinates
2
3
1
(x ,x ,x )= (r, θ, z), where x = r cos θ, y = r sin θ, z = z.
I 2. Find the nonzero Christoffel symbols of the first and second kind in spherical coordinates
1
2
3
(x ,x ,x )= (ρ, θ, φ), where x = ρ sin θ cos φ, y = ρ sin θ sin φ, z = ρ cos θ.
I 3. Find the nonzero Christoffel symbols of the first and second kind in parabolic cylindrical coordinates
1
1 2 3 2 2
(x ,x ,x )= (ξ, η, z), where x = ξη, y = (ξ − η ), z = z.
2
I 4. Find the nonzero Christoffel symbols of the first and second kind in parabolic coordinates
1 2 2
3
1
2
(x ,x ,x )= (ξ, η, φ), where x = ξη cos φ, y = ξη sin φ, z = (ξ − η ).
2
I 5. Find the nonzero Christoffel symbols of the first and second kind in elliptic cylindrical coordinates
2
3
1
(x ,x ,x )= (ξ, η, z), where x =cosh ξ cos η, y = sinh ξ sin η, z = z.
I 6. Find the nonzero Christoffel symbols of the first and second kind for the oblique cylindrical coordinates
2
3
1
(x ,x ,x )= (r, φ, η), where x = r cos φ, y = r sin φ+η cos α, z = η sin α with 0 <α < π and α constant.
2
Hint: See figure 1.3-18 and exercise 1.3, problem 12.
∂g ik
I 7. Show [ij, k]+ [kj, i]= .
∂x j
I 8.
r ri

(a) Let = g [st, i] and solve for the Christoffel symbol of the first kind in terms of the Christoffel
st
symbol of the second kind.

n
(b) Assume [st, i]= g ni and solve for the Christoffel symbol of the second kind in terms of the
st
Christoffel symbol of the first kind.
I 9.
(a) Write down the transformation law satisfied by the fourth order tensor ijk,m .
(b) Show that ijk,m = 0 in all coordinate systems.
√
(c) Show that ( g) ,k =0.

I 10. Show ijk =0.
,m
I 11. Calculate the second covariant derivative A i,kj .

∂φ
~ i
2
1
3
I 12. The gradient of a scalar field φ(x ,x ,x ) is the vector grad φ = E .
∂x i
(a) Find the physical components associated with the covariant components φ ,i
i
dφ A φ ,i
i
(b) Show the directional derivative of φ in a direction A is = .
n 1/2
dA (g mn A A )
m