We define
ι
v
β
=
β
(
v
)
∈
R
for a 1-covector
β
on
V
and
ι
v
β
=
0 for a 0-covector
β
(a
constant) on
V
.
Proposition 20.7.
For
1
-covectors
α
1
,...,
α
k
on a vector space V and v
∈
V,
ι
v
(
α
1
∧···∧
α
k
)=
k
∑
i
=
1
(
−
1
)
i
−
1
α
i
(
v
)
α
1
∧···∧
hatwide
α
i
∧···∧
α
k
,
where the caret
hatwide
over
α
i
means that
α
i
is omitted from the wedge product.
Proof.
parenleftBig
ι
v
parenleftBig
α
1
∧···∧
α
k
parenrightBigparenrightBig
(
v
2
,...,
v
k
)
=
parenleftBig
α
1
∧···∧
α
k
parenrightBig
(
v
,
v
2
,...,
v
k
)
=
det
α
1
(
v
)
α
1
(
v
2
)
···
α
1
(
v
k
)
α
2
(
v
)
α
2
(
v
2
)
···
α
2
(
v
k
)
.
.
.
.
.
.
.
.
.
.
.
.
α
k
(
v
)
α
k
(
v
2
)
···
α
k
(
v
k
)
(
Proposition 3.27
)
=
k
∑
i
=
1
(
−
1
)
i
+
1
α
i
(
v
)
det
[
α
ℓ
(
v
j
)]
1
≤
ℓ
≤
k
,ℓ
negationslash
=
i
2
≤
j
≤
k
(
expansion along first column
)
=
k
∑
i
=
1
(
−
1
)
i
+
1
α
i
(
v
)
parenleftBig
α
1
∧···∧
hatwide
α
i
∧···∧
α
k
parenrightBig
(
v
2
,...,
v
k
)
(Proposition 3.27)
.
⊓
⊔
Proposition 20.8.
For v in a vector space V, let
ι
v
:
logicalandtext
∗
(
V
∨
)
→
logicalandtext
∗−
1
(
V
∨
)
be interior
multiplication by v. Then
(i)
ι
v
◦
ι
v
=
0
,
(ii)
for
β
∈
logicalandtext
k
(
V
∨
)
and
γ
∈
logicalandtext
ℓ
(
V
∨
)
,
ι
v
(
β
∧
γ
)=(
ι
v
β
)
∧
γ
+(
−
1
)
k
β
∧
ι
v
γ
.
In other words,
ι
v
is an antiderivation of degree
−
1
whose square is zero.
Proof.
(i) Let
β
∈
logicalandtext
k
(
V
∨
)
. By the definition of interior multiplication,
(
ι
v
(
ι
v
β
))(
v
3
,...,
v
k
)=(
ι
v
β
)(
v
,
v
3
,...,
v
k
)=
β
(
v
,
v
,
v
3
,...,
v
k
)=
0
,
because
β
is alternating and there is a repeated variable
v
among its arguments.

228
§
20
The Lie Derivative and Interior Multiplication
(ii) Since both sides of the equation are linear in
β
and in
γ
, we may assume that
β
=
α
1
∧···∧
α
k
,
γ
=
α
k
+
1
∧···∧
α
k
+
ℓ
,
where the
α
i
are all 1-covectors. Then
ι
v
(
β
∧
γ
)
=
ι
v
(
α
1
∧···∧
α
k
+
ℓ
)
=
parenleftBigg
k
∑
i
=
1
(
−
1
)
i
−
1
α
i
(
v
)
α
1
∧···∧
hatwide
α
i
∧···∧
α
k
parenrightBigg
∧
α
k
+
1
∧···∧
α
k
+
ℓ
+(
−
1
)
k
α
1
∧···∧
α
k
∧
k
∑
i
=
1
(
−
1
)
i
+
1
α
k
+
i
(
v
)
α
k
+
1
∧···∧
hatwidest
α
k
+
i
∧···∧
α
k
+
ℓ
(by Proposition 20.7)
= (
ι
v
β
)
∧
γ
+(
−
1
)
k
β
∧
ι
v
γ
.
⊓
⊔
Interior multiplication on a manifold is defined pointwise. If
X
is a smooth vector
field on
M
and
ω
∈
Ω
k
(
M
)
, then
ι
X
ω
is the
(
k
−
1
)
-form defined by
(
ι
X
ω
)
p
=
ι
X
p
ω
p
for all
p
∈
M
. The form
ι
X
ω
on
M
is smooth because for any smooth vector fields
X
2
,...,
X
k
on
M
,
(
ι
X
ω
)(
X
2
,...,
X
k
)=
ω
(
X
,
X
2
,...,
X
k
)
is a smooth function on
M
(Proposition 18.7(iii)
⇒
(i)).
Of course,
ι
X
ω
=
ω
(
X
)
for a 1-form
ω
and
ι
X
f
=
0 for a function
f
on
M
. By the properties of interior
multiplication at each point
p
∈
M
(Proposition 20.8), the map
ι
X
:
Ω
∗
(
M
)
→
Ω
∗
(
M
)
is an antiderivation of degree
−
1 such that
ι
X
◦
ι
X
=
0.

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- Math, Algebra, Derivative, Manifold, Differential form, differential forms