PROBLEMSET6,PART2:INTEGRALSONMANIFOLDSDUE:DEC.14(1)[Orientable manifolds](a) Prove: Any Lie group G is orientable.(b) [not required] Suppose M is orientable, f : M -→ N is smooth, and q E N is aregular value of f.Prove: f-(q) is orientable.[Orientation-preserving diffeomorphism](2)(a) Characterize orientation-preserving diffeomorphism via volume forms.(b) Let f : Rn → Rn be the map f(r) = -r. Is f orientation-preserving?(c) Let f : sn → sn be the antipodal map f(a) = -r. Is f orientation-preserving?(d) Prove: For any k, RIPp2k is not orientable.(3) [An integral]Considerthe2-formw=rdy^dz+ydz^dr+zdadyOn R3. Let t : S2 → R3 be the standard embedding. Compute the integral Jsa 1*w.(4) [The divergence of a smooth vector field]Let M be a smooth manifold, μ a volume form on M, and X a smooth vector fieldon M. Then the Lie derivative Cxμ is again a smooth n-form on M. The divergenceof X with respect to the volume form μ is the function divμ(X) such thatLxμ= divμ(X)μ.(a) Find the expression of divμ(X) in local coordinates(b) Prove: For any f e Co(M),divμ(gX) = gdivμ(X) + X(g)μ.(c) Prove Gauss formula:(divμX)μ=ixu87A(d) We say a vector field X is divergence free if divμ(X) = 0. Prove: If M is acompact manifold without boundary (i.e. M = 0), and X is a divergence freevector field on M, then for any f,g E C(M), one hasx(f)gμ=-fx(g)μ(In other words, the linear operator X :Co(M)→ C(M) is skew-symmetric.)1
PROBLEM SET 6, PART 2: INTEGRALS ON MANIFOLDS DUE: DEC. 14 (1) [Orientable manifolds] (a) Prove: Any Lie group G is orientable. (b) [not required] Suppose M is orientable, f : M → N is smooth, and q ∈ N is a regular value of f. Prove: f −1 (q) is orientable. (2) [Orientation-preserving diffeomorphism] (a) Characterize orientation-preserving diffeomorphism via volume forms. (b) Let f : R n → R n be the map f(x) = −x. Is f orientation-preserving? (c) Let f : S n → S n be the antipodal map f(x) = −x. Is f orientation-preserving? (d) Prove: For any k, RP2k is not orientable. (3) [An integral] Consider the 2-form ω = xdy ∧ dz + y 2 dz ∧ dx + zdx ∧ dy on R 3 . Let ι : S 2 → R 3 be the standard embedding. Compute the integral R S2 ι ∗ω. (4) [The divergence of a smooth vector field] Let M be a smooth manifold, µ a volume form on M, and X a smooth vector field on M. Then the Lie derivative LXµ is again a smooth n-form on M. The divergence of X with respect to the volume form µ is the function divµ(X) such that LXµ = divµ(X)µ. (a) Find the expression of divµ(X) in local coordinates. (b) Prove: For any f ∈ C∞(M), divµ(gX) = gdivµ(X) + X(g)µ. (c) Prove Gauss formula: Z M (divµX)µ = Z ∂M ιXµ. (d) We say a vector field X is divergence free if divµ(X) = 0. Prove: If M is a compact manifold without boundary (i.e. ∂M = ∅), and X is a divergence free vector field on M, then for any f, g ∈ C∞(M), one has Z M X(f)gµ = − Z M fX(g)µ. (In other words, the linear operator X : C∞(M) → C∞(M) is skew-symmetric.) 1
2PROBLEMSET6,PART2:INTEGRALSONMANIFOLDSDUE:DEC.14(5)[The transport equation] [not required]Let M bea smoothmanifold, μavolumeform on M, and 2CM an open set inMwith smooth boundary. Let X be a complete vector field on M, and 2t = Φt(2) theflow-out of 2by the flowΦt generated by X(a) For any p E Co(R × M) (as usual we denote Pt = p(t,-) E Co(M), prove:d(% + div (ptx) μ.Ptμ=[The transport equation]J.（otdt Jat(b) Let p be the mass density of an ideal fuid moving in a compact region. Assumethe conservation of mass, i.e. for any region 2, Jo, Ptμ is independent of t. Prove:op+ divμ(ptX) = 0.[Theequationof continuity]t
2 PROBLEM SET 6, PART 2: INTEGRALS ON MANIFOLDS DUE: DEC. 14 (5) [The transport equation] [not required] Let M be a smooth manifold, µ a volume form on M, and Ω ⊂ M an open set in M with smooth boundary. Let X be a complete vector field on M, and Ωt = ϕt(Ω) the flow-out of Ω by the flow ϕt generated by X. (a) For any ρ ∈ C∞(R × M) (as usual we denote ρt = ρ(t, ·) ∈ C∞(M)), prove: Movingtoright d dt Z Ωt ρtµ = Z Ωt ∂ρ ∂t + divµ(ρtX) µ. [The transport equation] (b) Let ρ be the mass density of an ideal fluid moving in a compact region. Assume the conservation of mass, i.e. for any region Ω, R Ωt ρtµ is independent of t. Prove: Movingtoright∂ρ ∂t + divµ(ρtX) = 0. [The equation of continuity]