Multidimensional Calculus: Differential Theory
Chapter & Page: 9–19
9.5
General Formulas for the Divergence and Curl
Later, we will discover the geometric significance of the divergence and the curl of a vector field.
Then, using those, we can both redefine divergence and curl in a coordinate-free manner, and
obtain more general formulas for these. This will come from our development of the divergence
(or Gauss’s) theorem and the Stokes’ theorem. In the meantime, I will simply tell you what those
formulas are, and we will use them, as needed, with the understanding that, eventually, we will
see how to derive them.
As usual, let
braceleftbig(
x
1
,
x
2
,...,
x
N
)bracerightbig
be an
orthogonal
coordinate system for our space
S
with
{
h
1
,
h
2
,...,
h
N
}
,
{
ε
1
,
ε
2
,...,
ε
N
}
and
{
e
1
,
e
2
,...,
e
N
}
being the associated scaling factors, tangent vectors and normalized tangent vectors at each point.
Assume
F
is a vector field with component/coordinate formula
F
(
x
)
=
N
summationdisplay
i
=
1
F
i
(
x
1
,
x
2
,...,
x
N
)
e
i
where
x
∼
(
x
1
,
x
2
,...,
x
N
)
.
The corresponding formulas for the divergence of
F
depend somewhat on the dimension
N
of the space, though the pattern will become obvious. If
N
=
2 , then
∇
·
F
=
1
h
1
h
2
braceleftbigg
∂
∂
x
1
bracketleftbig
F
1
h
2
bracketrightbig
+
∂
∂
x
2
bracketleftbig
F
2
h
1
bracketrightbig
bracerightbigg
.
(9.16a)
If
N
=
3 , then
∇
·
F
=
1
h
1
h
2
h
3
braceleftbigg
∂
∂
x
1
bracketleftbig
F
1
h
2
h
3
bracketrightbig
+
∂
∂
x
2
bracketleftbig
F
2
h
1
h
3
bracketrightbig
+
∂
∂
x
3
bracketleftbig
F
3
h
1
h
2
bracketrightbig
bracerightbigg
.
(9.16b)
If
N
=
4 , then
∇
·
F
=
1
h
1
h
2
h
3
h
4
braceleftbigg
∂
∂
x
1
bracketleftbig
F
1
h
2
h
3
h
4
bracketrightbig
+
∂
∂
x
2
bracketleftbig
F
2
h
1
h
3
h
4
bracketrightbig
+
∂
∂
x
3
bracketleftbig
F
3
h
1
h
2
h
4
bracketrightbig
+
∂
∂
x
4
bracketleftbig
F
4
h
1
h
2
h
3
bracketrightbig
bracerightbigg
.
(9.16c)
And so on.
The curl still only makes sense if the dimension
N
is three. Using Stokes’ theorem, we will
version: 10/31/2011