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 LeviCivita symbols
1
, and
∇
=
d
+
A
is the LeviCivita connection of the metric
g
on
M
, given exterior derivative
d
and matrixvalued oneforms (
A
i
)
k
j
= Γ
k
ij
that define this connection
(more generally, for any connection, the oneform
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
3
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 '09
 FRANSPRETORIUS
 Work, General Relativity, ln gkk, Γk ij

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