Sample Final Exam
Unless specified otherwise, M, N, L are smooth manifolds of dimensions n, k, l respectively.
1. Prove that for any f ∈C∞(M) andv ∈TpM (p∈M) dfp(v) = vf.
2. InR3, find the dual basis of v1 = ∂
∂x1, v2 = ∂
∂x1 + ∂
∂x2, v3 = ∂
∂x1 + ∂
∂x2 + ∂
∂x3, answer in terms of {dx1, dx2, dx3}.
3. Express the following vector field in R2\ {(0,0)}:
X =a ∂
∂x +b ∂
∂y (a, b∈R) in the polar coordinates (r, θ) whose basis tangent vectors are ∂
∂r, ∂
∂θ. Hint. r=p
x2+y2, θ= arctany x
.
4. For the product manifold M ×N, what is the cotangent bundle T∗(M×N)?
5. Determine the cotangent bundle for the open submanifold U ⊂M. 6. Consider on R2: let
X = (x2+y) ∂
∂x + (y2+ 1) ∂
∂y, ω= (2xy+x2+ 1)dx+ (x2−y)dy, and let F :R3 →R2 be the map
(u, v, w)7→(u−v, v2+w).
Compute
(a) ω(X)(0,0).
(b) F∗ω.
7. Let F be a diffeomorphism on M and X, ω are, respectively, smooth vector field and smooth covector field on M. Prove that
iX(F∗ω) =F∗(if Xω).
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8. Find an example of bilinear function Φ :Rn×Rk →R which cannot be expressed as Φ = ω⊗η,
for some linear functions ω:Rn →Rand η :Rk→R. 9. On R3, express the tensor field
J =dx⊗ ∂
∂x +dy⊗ ∂
∂y +dz⊗ ∂
∂z in the system of spherical coordinates given by
x=ρcosϕcosθ, y=ρcosϕsinθ, z =ρsinϕ.
10. Compute the integral curves of the vector field on R3 given by X =y ∂
∂x +y ∂
∂y + 2 ∂
∂z. 11. For each t ∈R, consider the mapθt :R2 →R2 given by
(x, y)7→θt(x, y) = (xcost+ysint,−xsint+ycost).
(a) Prove that {θt} is a one-parameter group of transformations on R2. (b) Calculate the vector field generated by this flow.
12. LetT M be the tangent bundle over a differentiable manifoldM. Let θ:R×T M →T M defined by θ(t, X) = etX.
(a) Prove that θ is a one-parameter group of transformations of T M.
(b) Calculate the vector field Y generated by the flow θ.
13. Determine a basis for the tensor space T(2,3)(TpM) where p ∈ M in a local coordinate systems about p.
14. In a local coordinate system {xi}of M, determine the local component functions for the tensor product
S⊗T, where S ∈Γ(Tk,`M), T ∈Γ(Tr,sM).
15. For each of the following questions, determine if the statement is correct. If the statement is correct please give a supporting reasoning, otherwise, give a counter-example.
(a) Every flow on R2 is complete.
(b) Every flow on Sn is complete.
16. Prove that the alternating operator Alt on a vector space V is linear and Alt(ei1 ⊗ · · ·eik) = 1
k!
X
σ∈S(k)
sgn(σ)(eiσ(1) ⊗ · · · ⊗eiσ(k)).
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17. In R4 whose coordinate system is {(x, y, z, w)}, calculate
(exdx−3dy+dw)∧(5dx∧dy+ cos(xy)dz∧dw).
18. Prove that the product of orientable manifolds is orientable.
19. Prove that T M is orientable.
20. Given on R3, the differential 2-form
ω= (z−x2−xy)dx∧dy−dy∧dz−dz∧dx, compute
Z
D
i∗ω,
where i:D ,→R3 and D={(x, y, z)∈R3 :x2+y2 61, z= 0}.
21. Compute the integral
Z
S1
(x−y2)dx+x3dy.
Hint. S1 =∂B, whereB ={(x, y)∈R2 :x2+y2 61}.
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