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Next, we generalize the differential of a function as defined by equation (4.31a)
to a differential of an arbitrary differential form.
Definition.
Given a k-form
Â
w
=
w
dx
Ÿ
...
Ÿ
dx
i
...
i
i
i
1
k
1
k
1
£< < £
i
K
i
n
1
k
define a (k + 1)-form dw, called the differential of w, by
Â
d
w
=
d
w
Ÿ
dx
Ÿ
...
Ÿ
dx
i
...
i
i
i
1
k
1
k
1
£< < £
i
K
i
n
1
k
n
Â
Â
(
)
=
D
w
dx
ŸŸŸ
dx
...
dx
.
j
i
...
i
j
i
i
1
k
1
k
1
£< < £
i
K
i
n
j
=
1
1
k
The map wÆdw is called the differential operator d for differential forms.
Note that if we consider a function f : R n Æ R as a 0-form, then the new defini-
tion of the differential of f agrees with our earlier one.
4.9.4.
Theorem.
(1) If w and h are two k-forms, then d(w+h) = dw+dh.
(2) If w is an r-form and h is an s-form, then
rs
(
) =Ÿ+ () Ÿ
d
wh
Ÿ
d
wh
1
w h
d
.
(3) d(dw) = 0
(4) If f : R n
Æ R m is differentiable and w is a k-form, then f*(dw) = d(f*w).
Proof. For fact (2), check the formula first on the 1-forms dx i and their wedge
products. Fact (3) is proved by direct computation using formula (4.25b) that will
cause terms to cancel. Fact (4) is proved by induction on k. See [Spiv65].
One can show that Theorem 4.9.4(1)-(3) and equation (4.31a) can be considered
axioms for the differential operator d that define it uniquely.
We now know all the basic facts we need to know about differential forms and
are ready to move on to the important integration applications. Before we do though,
we finish this section with definitions of several well-known concepts.
Let F be a vector field on R n
Definition.
with component functions F i . The diver-
gence of F, denoted by div F, is defined by
n
 1
div F
=
D ii
.
i
=
If n = 3, then the curl of F, denoted by curl F, is the vector field on R 3 defined by
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