92
6 Tensor Analysis
(c) Refer back to the discussion in Example 4.2 and especially (4.10). Can
this f (ρ) be the metric of a 2-surface imbedded in Euclidean 3-space?
6.6 The Levi-Cevita epsilon occurs often in matrix and tensor theory; in n
dimensions it has n indices and is defined in terms of its indices as
αβγ ...τ =
0 if 2 indices are equal
±1 for even/odd permutations of 1 2 . . . n
The epsilon is not a tensor, but the object quantity e αβγ ...τ =
√
|g| αβγ ...τ is a
tensor. Show this. Hint: Express the determinant of a matrix using the epsilon.
6.7 In three dimensions show that the epsilon obeys
i jk imn = δ jm δ kn − δ jn δ km , sum over i.
6.8 Show that the covariant derivative of a coordinate basis vector may be written
as
∇ ∇
e n =
i
nk
e i ⊗ ˜
dx
k
.
6.9 Let us see how nicely the divergence and the invariant volume element fit
together. To do this consider in Euclidean 3-space the volume integral of the
divergence of a vector. Use the divergence expression (6.29) and the invariant
volume element expression (4.81) and see how the volume integral becomes
a surface integral, that is Gauss’s Theorem. Is it clear how useful this is for a
spherically symmetric system in spherical coordinates?
6.10 For some familiar important cases let’s solve Laplace’s equation, that is setting
the Laplacian of a scalar function equal to zero. Using the results of Exercise
6.3…
(a) Solve it for a spherically symmetric system in spherical coordinates.
Notice that you must allow a singularity or the only solution is zero!
This solution is useful in Part III.
(b) Solve it for a cylindrically symmetric system in cylindrical coordinates.
6 Tensor Analysis
(c) Refer back to the discussion in Example 4.2 and especially (4.10). Can
this f (ρ) be the metric of a 2-surface imbedded in Euclidean 3-space?
6.6 The Levi-Cevita epsilon occurs often in matrix and tensor theory; in n
dimensions it has n indices and is defined in terms of its indices as
αβγ ...τ =
0 if 2 indices are equal
±1 for even/odd permutations of 1 2 . . . n
The epsilon is not a tensor, but the object quantity e αβγ ...τ =
√
|g| αβγ ...τ is a
tensor. Show this. Hint: Express the determinant of a matrix using the epsilon.
6.7 In three dimensions show that the epsilon obeys
i jk imn = δ jm δ kn − δ jn δ km , sum over i.
6.8 Show that the covariant derivative of a coordinate basis vector may be written
as
∇ ∇
e n =
i
nk
e i ⊗ ˜
dx
k
.
6.9 Let us see how nicely the divergence and the invariant volume element fit
together. To do this consider in Euclidean 3-space the volume integral of the
divergence of a vector. Use the divergence expression (6.29) and the invariant
volume element expression (4.81) and see how the volume integral becomes
a surface integral, that is Gauss’s Theorem. Is it clear how useful this is for a
spherically symmetric system in spherical coordinates?
6.10 For some familiar important cases let’s solve Laplace’s equation, that is setting
the Laplacian of a scalar function equal to zero. Using the results of Exercise
6.3…
(a) Solve it for a spherically symmetric system in spherical coordinates.
Notice that you must allow a singularity or the only solution is zero!
This solution is useful in Part III.
(b) Solve it for a cylindrically symmetric system in cylindrical coordinates.