4.2. STOKES’ THEOREM 95
Writex′ = (x1, . . . , xn−1) ∈ Rn−1. By assumption the Jacobian matrix ofτ = (τ1, . . . , τn)at(x′,0)∈∂U has positive determinant and block form
A B C D
, where
C = ∂τn
∂x1(x′,0), . . . , ∂τn
∂xn−1(x′,0)
= (0, . . . ,0) sinceτn(x′,0) = 0for allx′, and
D= ∂τn
∂xn
(x′,0)>0
sinceτ maps the upper half space into itself. It follows thatA, which is the Jacobian of∂τat(x′,0), also has positive determinant, as desired.
LetMbe a smooth manifold with boundary. It follows from Lemma 4.2.1 that theboundary of M, namely, the subset∂M consisting of points of M mapped to∂Rn+ under coordinate charts, is well defined. Moreover, it is a smooth manifold of dimension(n−1), and an (oriented) atlas forM in-duces an (oriented) atlas for∂M by restricting the coordinate charts. Note also thatM\∂M is a smooth manifold of dimensionn.
4.2.2 Examples (a) The closed unit ballB¯ninRnis a smooth manifold with boundarySn−1.
(b) The Möbius band is smooth manifold with boundary a circleS1. LetM be a smooth manifold with boundary of dimensionn. The tan-gent space toMat a pointpis ann-dimensional vector space defined in the same way as in the case of a smooth manifold (even in casep ∈∂M). The definition of the tangent bundle also works, andT M is itself a manifold with boundary∂(T M) = ˙∪p∈∂MTpM. More generally, tensor bundles and differential forms are also defined. IfM is in addition oriented, the integral of compactly supportedn-forms is defined similarly to above.
In general, for an oriented smooth manifold with boundary, we will always use the so called induced orientation on its boundary. Namely, if inRn+ we use the standard orientation given bydx1 ∧ · · · ∧dxn, then the induced orientation on ∂Rn+ is specified by (−1)ndx1 ∧ · · · ∧ dxn−1 (the sign is required to make the statement of Stokes’ theorem right). On an oriented smooth manifold with boundaryM, for any local chart(U, ϕ)in an oriented atlas ofM, we declare the restriction ofϕto∂U → ∂Rn+ to be orientation-preserving.
A0-manifoldM is just a countable discrete collection of points. In this case, an orientation forM is an assignment of signσ(p) =±1for eachp ∈
4.2. STOKES’ THEOREM 97 M andR
Mf = P
p∈Mσ(p)f(p)for any0-formf ∈ C∞(M)with compact support.
As it is, the closed interval[a, b]⊂R(a < b) admits an orientation given by the nowhere vanishing1-formdx1, but no oriented atlas consisting of local charts with values onR1+! (Note that in the proof of Proposition 4.1.5, we used the fact that if(x1, . . . , xn)are local coordinates on our manifold, then so are(−x1, . . . , xn).) To remedy this situation, we introduce a slight modification in the definition of manifold with boundary in case n = 1 and also allow local charts with values on theleft-lineR1
−. Accordingly, for the standard orientation dx1 of R1−, the induced orientation is on∂Rn+ is specified by+1. With such conventions, the induced orientation atais−1 and that atbis+1.
4.2.3 Exercise Let M be an oriented n-manifold with boundary and give
∂M the induced orientation. Fixp ∈ ∂M. A vectorv ∈ TpM \Tp(∂M)is calledinward-pointing if v = ˙γ(0) for some smooth curveγ : [0, ǫ) → M with γ(0) = p; and it is called outward-pointing if −v is inward pointing.
Show that(v1, . . . , vn−1)is a basis that defines the orientation ofTp(∂M)if and only if(v, v1, . . . , vn−1) is a basis that defines the orientation ofTpM, wherev∈TpM is outward-pointing.
4.2.4 Remark A smooth manifoldMin the old sense is a smooth manifold with boundary with∂M = ∅. Indeed, we can always find an atlas forM whose local charts have images inRn+\∂Rn+.
Statement and proof of the theorem
4.2.5 Theorem Letω be an(n−1)-form with compact support on an oriented smoothn-manifoldMwith boundary and give∂M the induced orientation. Then
Z
M
dω= Z
∂M
ω.
In the right hand side of Stokes’ theorem, ω is viewed as ι∗ω, where ι:∂M → M is the inclusion, and the integral vanishes if∂M = ∅. In the casen= 1, the integral on the right hand side is a finite sum and the result reduces to the Fundamental Theorem of Calculus.
Proof of Theorem 4.2.5.We first consider two special cases.
Case 1: Mis an open subsetU ofRn. Viewωas an(n−1)-form onRn with compact support contained inU. Writeω=P
iaidx1∧ · · · ∧dxˆi∧ · · · ∧
dxn. Thendω=P
i(−1)i−1∂a∂xi
idx1∧ · · · ∧dxn. By Fubini’s theorem, Z
U
dω = Z
Rn
dω
= X
i
(−1)i−1 Z
Rn−1
Z ∞
−∞
∂ai
∂xidxi
dx1· · ·dxˆi· · ·dxn
= 0 because
Z ∞
−∞
∂ai
∂xidxi
= ai(. . . , xi−1,∞, xi+1, . . .)−ai(. . . , xi−1,−∞, xi+1, . . .)
= 0,
asai has compact support. SinceM has no boundary, this case is settled.
Case 2:Mis an open subsetU ofRn+. Viewωas an(n−1)-form onRn+ with compact support contained inU. Writeω =P
iaidx1∧ · · · ∧dxˆi∧ · · · ∧ dxnas before, but note that while theaiare smooth on (a neighborhood) of Rn+, the linear formsdxiare defined on the entireRn. Sinceaihas compact support,R∞
−∞
∂ai
∂xidxi = 0fori < n, so by Fubini’s theorem Z
U
dω = Z
Rn
+
dω
= (−1)n−1 Z
Rn−1
Z ∞ 0
∂an
∂xndxn
dx1· · ·dxn−1
= (−1)n−1 Z
Rn−1−an(x1, . . . , xn−1,0)dx1· · ·dxn−1
= Z
∂Rn
+
ω
= Z
∂U
ω,
where we have used thatdxn= 0onRn−1× {0}, finishing this case.
General case: M is an arbitrary manifold with boundary of dimen-sion n. Let{(Uα, ϕα)} be an oriented atlas for M such that each Uα has compact closure and let {ρα} be a partition of unity strictly subordinate to{Uα}. Thenω =P
αραωwhere each term has compact support. By lin-earity, it suffices to prove Stokes’ formula forραωwhich has support con-tained inUα. SinceUα is diffeomorhic to an open set inRnorRn
+, cases 1 and 2 imply that the formula holds onUα, so
Z
M
dραω= Z
Uα
dραω= Z
∂Uα
ραω = Z
∂M
ραω,
which concludes the proof of the theorem.
4.2. STOKES’ THEOREM 99 Manifolds with corners and Stokes’ theorem
The theory of smooth manifolds with corners is not a completely standard-ized theory and we just sketch a group of definitions which leads us quickly to a version of Stokes’ theorem.
Define then-sectorRn++to be the subset ofRndefined byx1≥0, . . . , xn≥ 0. A point p ∈ Rn
++ is called acorner of index k (0 ≤ k ≤ n) if exactlyk coordinates ofp vanish. The basic result the we need is contained in the following exercise.
4.2.6 Exercise Letτ : U → V be a diffeomorphism between open subsets ofRn
++. Thenp∈Rn
++is a corner of indexkif and only ifτ(p)is a corner of indexk.
Asmooth manifold with cornersM of dimension nis given by a smooth atlas where the local charts are homeomorphisms onto open subsets of Rn
++ and the transition maps are diffeomorphisms between open subsets ofRn++. Tensor bundles, differential forms, orientation and integration of forms are defined in the usual way.
Ann-manifold with cornersMis stratified by locally closed subsetsMk consisting of corners of indexk, that is, points corresponding under a chart to corners of indexkinRn++(this is a good definition due to Exercise 4.2.6).
In general, the locusM≤k of corners of index at mostk is an open subset of M and a manifold with corners. In particular, M≤1 is a manifold with boundary. For instance, if M = [0,1]2 is the unit square, then M≤1 isM with the four vertices removed.
We define theboundary∂M to be the locusM≥1 of corners of index at least1. Since the locusM≥2 of corners of index at least2is a closed subset of measure zero in∂M, it is immaterial for the calculation of integrals. Note that(Rn++)1 =∪ni=1Hi whereHi is defined byxi = 0andxj >0forj 6=i.
EachHiis oriented by the(n−1)-form(−1)idx1∧ · · · ∧dxci∧ · · · ∧dxn. For an oriented smooth manifold with cornersM, the induced orientation at a point inM1by definition corresponds under a local chart to the orientation at the corresponding point ofHifor somei.
Ifωbe a compactly supported(n−1)-form on ann-manifold with cor-nersM, then Stokes’ theoremR
Mdω =R
∂Mωholds, where the right-hand side is interpreted as the integral of the pull-back ofωtoM1. Indeed, for an oriented atlasA={(Uα, ϕα)}ofMwith partition of unity{ρi}subordinate to{Uα},suppρi ⊂Uα(i), we haveR
M1ω =P
i
R
M1∩Uα(i)ρiω, where the sum on the right-hand side is finite. Stokes’ formula in this case follows as in Theorem 4.2.5 by using the following result.
4.2.7 Exercise Let ω be a compactly supported (n−1)-form on an open subsetU ofRn++. ThenR
Udω=R
∂Uω.