- Source: Smooth coarea formula
In Riemannian geometry, the smooth coarea formulas relate integrals over the domain of certain mappings with integrals over their codomains.
Let
M
,
N
{\displaystyle \scriptstyle M,\,N}
be smooth Riemannian manifolds of respective dimensions
m
≥
n
{\displaystyle \scriptstyle m\,\geq \,n}
. Let
F
:
M
⟶
N
{\displaystyle \scriptstyle F:M\,\longrightarrow \,N}
be a smooth surjection such that the pushforward (differential) of
F
{\displaystyle \scriptstyle F}
is surjective almost everywhere. Let
φ
:
M
⟶
[
0
,
∞
)
{\displaystyle \scriptstyle \varphi :M\,\longrightarrow \,[0,\infty )}
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
{\displaystyle \int _{x\in M}\varphi (x)\,dM=\int _{y\in N}\int _{x\in F^{-1}(y)}\varphi (x){\frac {1}{N\!J\;F(x)}}\,dF^{-1}(y)\,dN}
∫
x
∈
M
φ
(
x
)
N
J
F
(
x
)
d
M
=
∫
y
∈
N
∫
x
∈
F
−
1
(
y
)
φ
(
x
)
d
F
−
1
(
y
)
d
N
{\displaystyle \int _{x\in M}\varphi (x)N\!J\;F(x)\,dM=\int _{y\in N}\int _{x\in F^{-1}(y)}\varphi (x)\,dF^{-1}(y)\,dN}
where
N
J
F
(
x
)
{\displaystyle \scriptstyle N\!J\;F(x)}
is the normal Jacobian of
F
{\displaystyle \scriptstyle 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
{\displaystyle \scriptstyle y\,\in \,N}
is a regular point of
F
{\displaystyle \scriptstyle F}
and hence the set
F
−
1
(
y
)
{\displaystyle \scriptstyle F^{-1}(y)}
is a Riemannian submanifold of
M
{\displaystyle \scriptstyle M}
, so the integrals in the right-hand side of the formulas above make sense.
References
Chavel, Isaac (2006) Riemannian Geometry. A Modern Introduction. Second Edition.