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Past exam of the mathematics course of the University of Cambridge
/
2019
/
ii
/
Paper 3
/
37D
/
a
/
iii
/
Solution
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2019
ii
Paper 3
37D
a
iii
Created
2026-09-24
Updated
2026-10-03
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V
=
∑
a
f
a
Z
(
a
)
and
W
=
∑
b
g
b
Z
(
b
)
. Relation (
2
) reduces
[
V
,
W
]
to
brackets
of individual summands. Applying relations (
3)
, (
1)
, and (
3)
, in that order, gives
[
f
a
Z
(
a
)
,
g
b
Z
(
b
)
]
=
−
g
b
Z
(
b
)
(
f
a
)
Z
(
a
)
−
f
a
g
b
[
Z
(
b
)
,
Z
(
a
)
]
+
f
a
Z
(
a
)
(
g
b
)
Z
(
b
)
.
(1)
The middle term vanishes because the
Z
(
a
)
commute, and every remaining term lies in their smooth span. Therefore
[
V
,
W
]
∈
span
C
∞
{
Z
(
1
)
,
…
,
Z
(
n
)
}
.
(2)
This is the involutivity condition in the
Frobenius theorem
.
Ancestors
(12)
iii
a
37D
Paper 3
ii
2019
Past exam of the mathematics course of the University of Cambridge
Mathematics course of the University of Cambridge
Course of the University of Cambridge
University of Cambridge
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