10 BETANCOURT ET AL u cQ (4)cR Q=SxR ucQ (lha)C R2 (a) (6) (c) FIG 5.(a)The cylinder,Q=Sx R.is a nontrivial erample of a manifold.Although not globally equivalent wherever the two neighborhoods intersect (the intersections here shoun in gray). equipped with a differential structure. fua:vahael' consisting of open neighborhoods in Q, ua cQ, and homeomorphic charts, a:la→yaCR”, that are smooth functions whenever their domains overlap(Figure 5), (R"),Va,B nug#0. Coordinates subordinate to a chart, q:4a→R where i is the ith Euclidean projection on the image of,provide local parameterizations of the manifold convenient for explicit calculations.10 BETANCOURT ET AL. Q = S 1 × R (a) U1 ⊂ Q U2 ⊂ Q (b) ψ1(U1) ⊂ R 2 ψ2(U2) ⊂ R 2 (c) Fig 5. (a) The cylinder, Q = S 1 ×R, is a nontrivial example of a manifold. Although not globally equivalent to a Euclidean space, (b) the cylinder can be covered in two neighborhoods (c) that are themselves isomorphic to an open neighborhood in R 2 . The manifold becomes smooth when ψ1 ◦ψ −1 2 : R 2 → R 2 is a smooth function wherever the two neighborhoods intersect (the intersections here shown in gray). equipped with a differential structure, {Uα, ψα}α∈I , consisting of open neighborhoods in Q, Uα ⊂ Q, and homeomorphic charts, ψα : Uα → Vα ⊂ R n , that are smooth functions whenever their domains overlap (Figure 5), ψβ ◦ ψ −1 α ∈ C ∞(R n ), ∀α, β | Uα ∩ Uβ 6= ∅. Coordinates subordinate to a chart, q i : Uα → R q → πi ◦ ψα, where πi is the ith Euclidean projection on the image of ψα, provide local parameterizations of the manifold convenient for explicit calculations