Integration on manifolds
Let be a finite-dimensional smooth manifold. Tangent spaces and such provide the global analogues of differential calculus. There is also an "integral calculus on manifolds" . Let
be the standard
-cube. A singular cube in
is a smooth mapping
. Let
be a
-form on
(cf. Differential form). Then the integral of
over a singular
-cube
is defined as
![]() | (a1) |
where is the unique smooth function such that
on
and where on the right-hand side the ordinary Lebesgue integral is taken. A singular
-chain is a formal finite sum
of singular
-cubes with coefficients in
. One defines
![]() | (a2) |
Now let be oriented and let
and
be two singular
-chains such that
for all
and such that all the
are orientation preserving. Then
. In particular, if the
fit together to define a piecewise-smooth
-dimensional submanifold
of
, then the integral
is well-defined.
Let denote the exterior derivative on exterior forms (cf. Exterior form) and
the (obvious) boundary operator on oriented (singular) chains. Then one has Stokes' theorem
![]() | (a3) |
where is a
-form and
is a singular
-chain. This is the analogue of the fundamental theorem of calculus.
A particular consequence is Green's theorem: Let be a compact
-dimensional manifold with boundary and let
be differentiable. Then
![]() | (a4) |
Let now be an oriented
-dimensional Riemannian manifold, i.e. for each
an orientation has been given on
. The volume form
on
is now defined by requiring that
for one (and hence each) orthonormal basis of
in the given orientation class of
. Another consequence of the general Stokes' theorem (a3) is the divergence theorem:
![]() | (a5) |
Here is a vector field on
,
is a three-dimensional oriented manifold in
,
if
,
is an outward normal to
, and
and
are, respectively, the volume and area elements of
and
. The inner product is induced from the standard one in
.
Finally there is the classical Stokes' formula: Let be an oriented two-dimensional submanifold with boundary
. Give
an orientation such that together with the outward normal it gives back the orientation of
. Let
parametrize
and let
be the vector field on
such that
everywhere. One then has the formula
![]() | (a6) |
where the curl of a vector field on
is defined by:
![]() | (a7) |
![]() |
All these theorems have higher-dimensional analogues.
References
[a1] | M. Spivak, "Calculus on manifolds" , Benjamin (1965) |
[a2] | M. Hazewinkel, "A tutorial introduction to differentiable manifolds and calculus on manifolds" W. Schiehlen (ed.) W. Wedig (ed.) , Analysis and estimation of stochastic mechanical systems , Springer (Wien) (1988) pp. 316–340 |
Integration on manifolds. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integration_on_manifolds&oldid=11797