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Past exam of the mathematics course of the University of Cambridge
/
2021
/
iii
/
Paper 303
/
3
/
b
/
Solution
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Past exam of the mathematics course of the University of Cambridge
2021
iii
Paper 303
3
b
2026-09-28
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Choose the positive local branch of the constraint,
σ
=
1
−
π
2
.
(1)
The
chain rule
gives
∂
i
σ
=
−
1
−
π
2
π
a
∂
i
π
a
,
(2)
and therefore
(
∂
i
n
)
2
=
(
∂
i
π
a
)
(
∂
i
π
a
)
+
1
−
π
2
π
a
(
∂
i
π
a
)
π
b
(
∂
i
π
b
)
.
(3)
Substitution into the original
free energy
gives exactly
F
[
π
]
=
∫
d
d
x
2
g
1
[
(
∂
i
π
a
)
(
∂
i
π
a
)
+
1
−
π
2
π
a
(
∂
i
π
a
)
π
b
(
∂
i
π
b
)
]
.
(4)
Ancestors
(11)
b
3
Paper 303
iii
2021
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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