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By Zhang B.

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2 Partitions of unity Definition 22 A partition of unity on M is a collection {ϕi }i∈I of smooth functions such that • ϕi ≥ 0 37 • {supp ϕi : i ∈ I} is locally finite • i ϕi = 1 Here locally finite means that for each x ∈ M there is a neighbourhood U which intersects only finitely many supports supp ϕi . 1 Given any open covering {Vα } of a manifold M there exists a partition of unity {ϕi } on M such that supp ϕi ⊂ Vα(i) for some α(i). We say that such a partition of unity is subordinate to the given covering.

0 ∂yn /∂xn (16) From the definition of manifold with boundary, ϕβ ϕ−1 α maps xn > 0 to yn > 0, so yn has the property that if xn = 0, yn = 0 and if xn > 0, yn > 0. It follows that ∂yn ∂xn > 0. xn =0 From (16) the determinant of the Jacobian for ∂M is given by det(J∂M ) ∂yn ∂xn = det(JM ) xn =0 so if det(JM ) > 0 so is det(J∂M ). ✷ Remark: The boundary of an oriented manifold has an induced orientation, but there is a convention about which one to choose: for a surface in R3 this is the choice of an “inward” or “outward” normal.

Now suppose Y is a vector field, considered as a map Y : M → T M . With a diffeomorphism F : M → M , its derivative DFx : Tx → TF (x) gives DFx (Yx ) ∈ TF (x) . This defines a new vector field Y˜ by Y˜F (x) = DFx (Yx ) (6) Thus for a function f , (Y˜ )(f ◦ F ) = (Y f ) ◦ F (7) Now if F = ϕt for a one-parameter group, we have Y˜t and we can differentiate to get ∂ ˜ Y˙ = Yt ∂t t=0 From (7) this gives Y˙ f + Y (Xf ) = XY f so that Y˙ = XY − Y X is the Lie derivative defined above. Thus the natural action of diffeomorphisms on vector fields specializes through one-parameter groups to the Lie bracket [X, Y ].

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2 T -periodic solution for m order neutral type differential equations with time delays by Zhang B.


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