ps3 - Physics 523 Introduction to Relativity Homework 3 Due...

Physics 523, Introduction to Relativity
Homework 3
Due Tuesday, 18
th
October 2011
Hans Bantilan
Gradient, Divergence, and Curl
Let
∂
∂x i
be the coordinate basis of T
p
M associated with coordinate chart x : U → Ω, where U ⊆ M is
a neighborhood containing p ∈ M , and Ω ⊂ IR
d
is open. For a scalar field f : M → IR , we may define its
gradient as
grad f := g
ij
∂f
∂x i
∂
∂x j
.
The divergence of a vector field Z = Z
i ∂
∂x i
is defined as
div Z :=
1
√
g
∂
∂x i
(
√
gZ
i
)
and its curl is defined as
curl Z :=
1
√
g
ijk
∇
j
Z
k
∂
∂x i
where
ijk
are the Levi-Civita symbols
1
, and ∇ = d + A is the Levi-Civita connection of the metric g
on M , given exterior derivative d and matrix-valued one-forms ( A
i
)
k
j
= Γ
k
ij
that define this connection
(more generally, for any connection, the one-form A that defines it is sometimes called the gauge field).
We are to consider these objects evaluated on IR
3
in spherical x i = { r, θ, φ } , with the flat metric then
given by
g = dr
2
+ r
2
dθ
2
+ r
2
sin
2
θdφ
2
.
To emphasize the simplicity of this calculation, we display it in its most elementary form by using
the coordinate basis
∂
∂r
,
∂
∂θ
,
∂
∂φ
(though this will lead us to expressions that are not immediately
comparable to those found using the orthonormal basis
∂
∂r
,
1
r
∂
∂θ
,
1
r sin θ
∂
∂φ
, for example). Computing
explicitly at a point p ∈ IR
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