In Riemannian geometry, the smooth coarea formulas relate integrals over the domain of certain mappings with integrals over their codomains.
Let
M
,
N
be smooth Riemannian manifolds of respective dimensions
m
≥
n
. Let
F
:
M
⟶
N
be a smooth surjection such that the pushforward (differential) of
F
is surjective almost everywhere. Let
φ
:
M
⟶
[
0
,
∞
]
a measurable function. Then, the following two equalities hold:
∫
x
∈
M
φ
(
x
)
d
M
=
∫
y
∈
N
∫
x
∈
F
−
1
(
y
)
φ
(
x
)
1
N
J
F
(
x
)
d
F
−
1
(
y
)
d
N
∫
x
∈
M
φ
(
x
)
N
J
F
(
x
)
d
M
=
∫
y
∈
N
∫
x
∈
F
−
1
(
y
)
φ
(
x
)
d
F
−
1
(
y
)
d
N
where
N
J
F
(
x
)
is the normal Jacobian of
F
, i.e. the determinant of the derivative restricted to the orthogonal complement of its kernel.
Note that from Sard's lemma almost every point
y
∈
N
is a regular point of
F
and hence the set
F
−
1
(
y
)
is a Riemannian submanifold of
M
, so the integrals in the right-hand side of the formulas above make sense.
