Dr. Wang Xingbo Fall ， 2005

1. Mathematical & Mechanical Method in Mechanical Engineering Dr. Wang Xingbo Fall，2005

2. Mathematical & Mechanical Method in Mechanical Engineering An Introduction to Manifolds • Review of Topology • Concepts of manifolds

3. Mathematical & Mechanical Method in Mechanical Engineering An Introduction to Manifolds How can we describe it?

4. Mathematical & Mechanical Method in Mechanical Engineering An Introduction to Manifolds • Many engineering objects have a shape of complicated surface. These complicated surfaces can be described by manifolds. Theories of manifolds have exhibited their elegances and excellences in many aspects of engineering, e.g. in controlling of robots, in structural analysis of mechanical engineering. Manifold is regarded to be a powerful tool for a senior engineer or a researcher to master.

5. Mathematical & Mechanical Method in Mechanical Engineering Review of Topology • A topological space is a pair (X, T) • where X is a set and T is a class of • subsets of X, called topology, which • satisfies the following three properties. • (i) X, ∈T. • (ii) If { X i}i∈I∈T, then ∪i∈I X i∈T • (iii) If X 1,…, X n∈T, then ∩i=1,…,n X i∈T.

6. Mathematical & Mechanical Method in Mechanical Engineering Review of Topology If (X,T) is a topological space, the elements of T are said open sets. A subset K of X is said closed if its complementary set X \K is open The closure of a set U X is the intersection of all the closed sets K X with U K

7. Mathematical & Mechanical Method in Mechanical Engineering Closure of a set

8. Mathematical & Mechanical Method in Mechanical Engineering Continuous function If (X,T) and (Y,U) are topological spaces, a mapping f : X→Y is said continuous if is open for each T∈U

9. Mathematical & Mechanical Method in Mechanical Engineering Homeomorphism An injective, surjective and continuous mapping f : X→Y, whose inverse mapping is also continuous, is said homeomorphism from X to Y. If there is a homeomorphism from X to Y these topological spaces are said homeomorphic.

10. Mathematical & Mechanical Method in Mechanical Engineering Base, second countable If (X,T) is a topological space, a class B T is said base of the topology, if each open set turns out to be union of elements of B. A topological space which admits a countable base of its topology is said second countable. If (X,T) is second countable, from any base B it is possible to extract a subbase B’B which is countable.

11. Mathematical & Mechanical Method in Mechanical Engineering Topology generated by set-class • If A is a class of subsets of X≠; and CA is • the class of topologies T on X with AT, • TA := T∈CA T is said the topology generated • by A. Notice that CA≠because the set • of parts of X, P(X), is a topology and • includes A.

12. Mathematical & Mechanical Method in Mechanical Engineering topology induced on a set If AX, where (X,T) is a topological space, the pair (A,TA) where, TA := {UA | UT},defines a topology on A which is said the topology induced on A by X.

13. Mathematical & mechanical Method in Mechanical Engineering Neighborhood • If (X,T) is a topological space and pX, a neighborhood of p is an open set UX with p∈U. If X and Y are topological spaces and xX, f: X→Y is said to be continuous in X, if for every neighborhood of f(x), VY , there is a neighborhood of x, UX, such that f(U) V . It is simply proven that f : X→Y as above is continuous if and only if it is continuous in every point of X.

14. Mathematical & mechanical Method in Mechanical Engineering Connect • A topological space (X,T) is said connected if there are no open sets A, B≠with AB =and AB = X. It turns out that if f: X→Y is continuous and the topological space X is connected, then f(Y) is a connected topological space when equipped with the topology induced by the topological space Y.

15. Mathematical & mechanical Method in Mechanical Engineering Hausdorff • A topological space (X,T) is said Hausdorff if each pair (p,q)XX admits a pair of neighborhoods Up, Uq with p∈Up, q∈Uq and UpUq=. If X is Hausdorff and xX is a limit of the sequence {Xn}n∈NX, this limit is unique.

16. Mathematical & mechanical Method in Mechanical Engineering Semi-distance A semi metric space is a set X endowed with a semidistance. d: XX→[0,+∞], with

17. Mathematical & mechanical Method in Mechanical Engineering Open Ball • The semidistance is called distance and the semi metric space is called metric space. • An open metric balls are defined as

18. Mathematical & mechanical Method in Mechanical Engineering Connected by path • A topological space (X,T) is said connected by paths if, for each pair p, qX there is a continuous path : [0,1] →X such that(0) = p,(1) = q,

19. Mathematical & mechanical Method in Mechanical Engineering Cover If X is any set, a covering of X is a class {Xi}i∈I, XiX for all iI, such that i∈I Xi = X

20. Mathematical & mechanical Method in Mechanical Engineering Compactness-Finite Cover A topological space (X,T) is said compact if from each covering of X, {Xi}i∈I are made of open sets, it is possible to extract a covering {Xj}j∈I of X with j finite. This is also called a finite covering property

21. Mathematical & mechanical Method in Mechanical Engineering Group • Let G be a set and  be a operation defined on W. If W and  satisfy the following regulations: • There is a unit e in G such that • . • where • Then G is called a group over R

22. Mathematical & mechanical Method in Mechanical Engineering isomorphism • Let S and T be tow groups with operations ,  respectively. If there exists a one-to-one mapping : ST such that, for any • 1. If it results in • 2. If are unit in S and T respectively, then • then S is said to be isomorphic to T, or vice versa; • the mapping  is said to be a isomorphism between S and T. • Two isomorphic groups can be regarded to have the same • structure algebraically

23. Mathematical & mechanical Method in Mechanical Engineering Concepts of Manifolds A topological space (X, T) is said topological manifold of dimension n if X is Hausdorff, second countable and is locally homeomorphic to Rn, i.e., for every pX there is a neighborhood pUp and a homomorphism p: Up→Vp where VpRn is a open set.

24. Mathematical & mechanical Method in Mechanical Engineering Chart (n-chart)Let X be topological space, U is an open subset of X. Let  be a homeomorphism from UX to an open subset V Rn, namely, : p→(x1(p),…,xn(p)). Then the ordered pair (U, )= C is called an n-chart on M. where Rn is the n-dimensional Euclidean space. A chart can be thought of a mapping from some open set to an open subset of Rn

25. Mathematical & mechanical Method in Mechanical Engineering Chart

26. Mathematical & mechanical Method in Mechanical Engineering K-Compatible • Let (U, ) and (U, ) be two charts on a topological space M . If UU, let V and V be image of UU under corresponding homeomorphisms  and . The two charts are said to be compatible if -1 viewed as a mapping from V Rn to V Rn, is a C function. If UU= then the charts are also said to be compatible. If -1 and -1 are all Ck (k<) functions, then  and  are said to Ck-compatible. If any  and  are said to C-compatible, then M is said to be smooth.

27. Mathematical & mechanical Method in Mechanical Engineering k-Compatible

28. Mathematical & Mechanical Method in Mechanical Engineering Atlas • An atlasA on a topological space M is a collection of charts{C} on M such that • Any two charts in atlas are piecewise k-compatible; • A covers M, i.e.

29. Mathematical & Mechanical Method in Mechanical Engineering Differential structure, Differentian Manifolds A differential structure on a topological space is an atlas with the property that any chart that is compatible with the charts of the atlas is also an element of the atlas. An n-dimensional differential manifoldM is a topological space endowed with a differential structure of n-charts.

30. Mathematical & Mechanical Method in Mechanical Engineering Local Cooridnate Systems If M is a n-dimensional differential manifold, then any point PM has such a open neighborhood U that is homeomorphism to an open set V of Rn, or we can say, that there exists at least one open subset U of M that has a n-chart (U,) such that (P)=VRn. At this time, the coordinate ((P))iof image(P) corresponding to P is called coordinate of PU and is denoted by xi(P)=((P))i. (U, xi)is called a local coordinate system.

31. Mathematical & Mechanical Method in Mechanical Engineering Local Cooridnate Systems It can be seen that, two charts (U,), (V,) on an n-dimensional differential manifold M are related two local coordinate systems. If UV，then there also exist two local coordinate systems corresponding to UV. Thus any point P UV has two coordinate representations xi(P)=((P))i and yi(P)=( (P))i and the two are dependent.

32. Mathematical & Mechanical Method in Mechanical Engineering Local Cooridnate Systems

33. Mathematical & Mechanical Method in Mechanical Engineering Differentiable Partitions of Unity on Manifolds

34. Mathematical & Mechanical Method in Mechanical Engineering Tensor Fields in Manifolds and Associated Geometric Structures Tangent , Tangent Bundle and State Space . Local representative Let be a continuous function from R to a differential manifold M.

35. Mathematical & Mechanical Method in Mechanical Engineering Tangent , Tangent Bundle and State Space .Local representative

36. Mathematical & Mechanical Method in Mechanical Engineering Related Two curves f and g are said to be related at p if and only if 1. f(0)=g(0)=p; 2.The derivatives of the local representations of f and g are equal

37. Mathematical & Mechanical Method in Mechanical Engineering Related properties If f(t) and g(t) are related in chart (U, ), they are also related in chart (V, )

38. Mathematical & Mechanical Method in Mechanical Engineering Related properties f(t) and g(t) are related in chart (U, )

39. Mathematical & Mechanical Method in Mechanical Engineering Related properties

40. Mathematical & Mechanical Method in Mechanical Engineering Related Properties

41. Mathematical & Mechanical Method in Mechanical Engineering Tangent space If M is a differentiable manifold and pM, the tangent space at point p, denoted as TpM, is defined to be the set of all equivalent classes Qp at p in M. TpM has the same dimension as M Define a map is injective

42. Mathematical & Mechanical Method in Mechanical Engineering Tangent space For any v in Rn, choose  such that for any |t|< is a path through in and is a smooth path through p is bijective , a linear isomorphic map from TpM to Rn

43. Mathematical & Mechanical Method in Mechanical Engineering Basis induced by a chart Let M be a differentiable manifold, pM, and take a chart (U,) with pU. If E1,…,En is the canonical basis of Rn, then define a basis in TpM which we call the basis induced in TpM by the chart (U,) (U,), (V, ) with pU,V and induced basis onTpM

44. Mathematical & Mechanical Method in Mechanical Engineering Derivations Symbol D(M) indicates the real vector space of all differential functions from manifold M to R indicate the vector space spanned by Let M be a differentiable manifold. A derivation in TpM is a R-linear map Dp: D(M)→R, such that, for each pair f, gD(M): Symbol DpM is used to indicate the R-vector space of the derivations in p

45. Mathematical & Mechanical Method in Mechanical Engineering Derivation Let M be a differential manifold. Take any TpM and any DpDpM (1) If hD(M) vanishes in a open neighborhood of p or, more strongly, h = 0 in the whole manifold M,then Dph= 0 (2) For every f, gD(M), Dpf = Dpg provide f(q) = g(q) in an open neighborhood of p.

46. Mathematical & Mechanical Method in Mechanical Engineering Flander's Lemma If f: B→R is C∞(B) where BRn is an open starshaped neighborhood of , then there are n differentiable mappings gi: B→R such that, if , then

47. Mathematical & Mechanical Method in Mechanical Engineering Flander’s lemma

48. Mathematical & Mechanical Method in Mechanical Engineering basis of TpM Let M be a differentiable manifold and pM. There exists a R-value vector space isomorphism F: TpMDpM such that, if is the basis of TpM induced by any local coordinate system about p with coordinates (x1,..., xn), it holds: And in particular the set is a basis of DpM

49. Mathematical & Mechanical Method in Mechanical Engineering Tangent Bundle The tangent bundle of a manifold M, denoted by TM is defined as the union of the tangent spaces for all pM. That is:

50. Mathematical & Mechanical Method in Mechanical Engineering Tangent Bundle TM is itself a differential manifold of dimension 2n TM= {(p, v) |pM , vTpM} Tangent bundle is called a state space