Sja Ch 9 meta notes

Sjamaar Chapter 9 Meta Notes PhL 8.23.15 Chapter is 9 pages total include Ex, this meta is 3 pages long. 9.1 Manifold with boundary 1 9.2 Integration over orientable manifolds 2 9.3 Gauss' and Stokes' classical theorems 2 9.1 Manifold with boundary Up to this point, a manifold was always described with embedding ψ(U) = V M where the domain U had to be an open set. Here that requirement is softened in a particular way. A special U' = U Hn allows your set U' to have a boundary along xn = 0 where n is the highest coordinate. If this creates a boundary in U', then the boundary M will also have a boundary. I suspect the idea really applies to any U which has any kind of reasonable boundary. The resulting M is called "a manifold with boundary" and the boundary is called ∂M of course. Then M-∂M ≡ int(M) is the interior. Example 9.2 is quite long in my raw notes. It is a particular ψ which results in M being a region in R2 which is bounded by a curve, as shown page 104. I think this is the first real example of a manifold which has the same dimensionality as the target space (the large space). Example 9.3 is an enhanced version of the previous example where you end up with a manifold which is a region of Rn which is bounded by a curved hypersurface. Theorem 9.4 extends the regular value theorem RVT φ(x) = c idea to a case where φm ≤ cm for I guess one of the φ components (instead of =). Obviously now φm = cm will produce ∂M instead of M, so Properties of ∂M dim(∂M) = N-m codimension(∂M) = N - [N-m] = m // original RVT Properties of M dim(M) = N-m+1 codimension(∂M) = N - [N-m+1] = m-1 2.15.16. In Ch 6 we had c = φ(x) being m equations with x in RN and this defined a surface of dimension N - m. In the above, we still have m equations but for the boundary, so it has dim = N-m. The inequality applies to M. Example 9.5 With φ(x) = ||x|| and φ = c we get a spherical surface as M (radius c). But then φ ≤ c causes that surface to be ∂M and then M is the solid sphere interior. Fact: This shows that a closed sphere of positive radius is a manifold with boundary, and the boundary in this case is the surface. For the sphere case if you had r1 ≤ φ(x) = ||x|| ≤ r2 where now both inequalities are on φ(x) and not on coordinates, then you get a thick spherical shell and it is a manifold, and ∂M is the union of the two spherical surfaces. No sharp corners, things are happy. Pair of pants is a manifold M, three closed curves comprise ∂M. No sharp corners. Mobius is a manifold, but not orientable because only has one side, a topic Sja did not really explore earlier or here. Tangent space for ∂M ? If you arrange to have [n, e1, e2.....en-1] be a TxM basis where n is normal to the assumed hypersurface, then [e1, e2.....en-1] is a basis for ∂M. If this IS the ∂M basis, then the M basis will be [n, e1, e2.....en-1] = (-1)n [e1, e2.....en] and so the "orientation" sign (+ or -) will depend on the integer n. You are inducing an orientation on ∂M from M is how he says it. 9.2 Integration over orientable manifolds I comment on cotangent space. In Lemma 9.6 Sja pauses to show that if two mappings c and c' have the same image M and have the same orientation preserving property as well, then integrals ∫[0,1]n c*α and ∫[0,1]n c'*α are the same. I think this would just be a reparametrization situation (with same orientation however). Now here is a major claim without proof: It is possible to represent any n-manifold M as the union of a finite set of mappings ci (each of an n-cube) such that these separate mappings have no overlap on M! Thus we write M = i=1k ci([0,1]n) where we assume k mappings are required Each of these ci mappings is orientation-preserving. Given this theorem, we can then say ∫M α = Σi=1k ∫[0,1]n ci*α = Σi=1k ∫ ΣI fI(ci(x)) det(Dci) dt1dt2......dtn // p 107 B Except for the right expression, this is page 107 B. This then is a prescription for doing an integration of any n-form α over any n-manifold M!! So we are combining Chapter 3 on pullbacks with Chapter 5 on Stokes and general pullbacks and "boundaries" ∂c and with Chap 6 on manifolds, and Chap 7 on differential forms on manifolds. Sja points out that the no-intersection requirement removes double counting in overlap regions when doing an integral. Here are a few sample integral types using the volume measure described above: (the Buck integrals!) volume of M = ∫M μM n=3 volume, n=2 surface area, n=1 arc length integral of f on M = ∫M μM f I comment a bit in the raw notes about the connection to Buck, but that topic is really not yet resolved fully. I will do it eventually. 9.3 Gauss' and Stokes' classical theorems One major result now is that Stokes' Theorem can be applied to an "orientable manifold with boundary": ∫M dα = ∫∂M α // page 108 A This section ends with a statement of special cases of this theorem: M = a 1-manifold: ∫M gdx = g(c(1))-g(c(0)) = g(b) - g(a) "other theorem" F = g ∫M F dx = g(c(1))-g(c(0)) = g(b) - g(a) "other theorem" F = f p 108 B This one is not mentioned in Chapter 9 ∫M (∂xg-∂yf) dxdy =∫∂M (fdx+gdy) ∂c = a curve "Green's theorem" M = an n-manifold: ∫M [div F] dx1dx2.....dxn = ∫∂M (F n) μ∂M divergence theorem (Gauss's theorem) p 108 D ∫M [curl (F n)] μM = ∫∂M Fdx Stokes's theorem p 109 A Notice in the second last line that we have μδM which is the volume form on the boundary. For an n→n mapping, you get = A = Jacobian, so then φ*dμ = Jacobian * dt1dt2.....dtn