Dr. Wang Xingbo Fall ， 2005

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

2. Mathematical & Mechanical Method in Mechanical Engineering Coordinate Systems Einstein Conventions k changes from 1 to n in an n-dimensional space

3. Mathematical & Mechanical Method in Mechanical Engineering Coordinate Systems Einstein Conventions

4. Mathematical & Mechanical Method in Mechanical Engineering Curvilinear coordinate systems change the value of , we will get a series of “parallel surfaces” in the space and for each point in the space, there exists one and only one surface passing through it.

5. Mathematical & Mechanical Method in Mechanical Engineering Curvilinear coordinate systems Take three different surfaces have a unique intersection point M,

6. Mathematical & Mechanical Method in Mechanical Engineering Curvilinear coordinate systems the two ordered triples of numbers are all coordinates of point M because both of them determine M uniquely in the space We call the ordered triples of numbers curvilinear coordinate of M.

7. Mathematical & mechanical Method in Mechanical Engineering Curvilinear Coordinate Systems Coordinate Transformation is Cartesian coordinate System then is a curvilinear coordinate System

8. Mathematical & mechanical Method in Mechanical Engineering Curvilinear Coordinate Systems i-th coordinate surface passing through P0 ui-line: ui-changge while the other two remain constant

9. Mathematical & mechanical Method in Mechanical Engineering Curvilinear Coordinate Systems Local basis

10. Mathematical & mechanical Method in Mechanical Engineering Curvilinear Coordinate Systems Notice Therefore, the local basis exist.

11. Mathematical & mechanical Method in Mechanical Engineering Curvilinear Coordinate Systems Arc-length

12. Mathematical & mechanical Method in Mechanical Engineering Curvilinear Coordinate Systems Normalized Basis Gradient of a Scalar Field F

13. Mathematical & mechanical Method in Mechanical Engineering Curvilinear Coordinate Systems Divergence of vector Field Curl of vector Field

14. Mathematical & mechanical Method in Mechanical Engineering Curvilinear Coordinate Systems Two Important things To be considered • two • Jacobian matrix is the fundamental of the transformation. If the matrix is regular, it means the transformation is invertible; otherwise the transformation is not invertible. • The metric in the new frame is important because it determine many metric properties such as distance, area, etc.

15. Mathematical & mechanical Method in Mechanical Engineering A Further Investigation on Local Coordinate System Suppose we have a complicated geometry object M a small part

16. Mathematical & Mechanical Method in Mechanical Engineering A Further Investigation on Local Coordinate System  be described by a differential function lay a smooth parameterized curve on 

17. Mathematical & Mechanical Method in Mechanical Engineering A Further Investigation on Local Coordinate System directional derivative of  along the curve

18. Mathematical & Mechanical Method in Mechanical Engineering A Further Investigation on Local Coordinate System many differential functions like , etc. to describe 

19. Mathematical & Mechanical Method in Mechanical Engineering A Further Investigation on Local Coordinate System consider two curves passing through the same point P

20. Mathematical & Mechanical Method in Mechanical Engineering A Further Investigation on Local Coordinate System the linear combination there must exist a curve

21. Mathematical & Mechanical Method in Mechanical Engineering A Further Investigation on Local Coordinate System Linear space: Tangent Space TP

22. Mathematical & Mechanical Method in Mechanical Engineering A Further Investigation on Local Coordinate System Basis vectors of coordinate system

23. Mathematical & Mechanical Method in Mechanical Engineering A Further Investigation on Local Coordinate System A surface with its tangent space is an impersonally in existence while we can subjectively illustrate them by many functions . There exists a function  such that are unit tangent vectors no matter what the actual form of  is . is a basis of the “ideal” state

24. Mathematical & Mechanical Method in Mechanical Engineering A Further Investigation on Local Coordinate System The tangent space has infinite frames at a point P but the frame with an “ideal” basis can simplify problem. That is why people usually use the ideal basis in theoretical research. However, we cannot have an ideal basis in engineering practice. This is way is of importance. In later parts of this book, we will, by convention, use the ideal basis for theoretical study.

25. Mathematical & Mechanical Method in Mechanical Engineering Transformation Between Basis and Coordinates

26. Mathematical & Mechanical Method in Mechanical Engineering Transformation Between Basis and Coordinates A vector X in 

27. Mathematical & Mechanical Method in Mechanical Engineering Transformation Between Basis and Coordinates

28. Mathematical & Mechanical Method in Mechanical Engineering Curvilinear Coordinate Systems

29. Mathematical & Mechanical Method in Mechanical Engineering Curvilinear Coordinate Systems A Surfacce

30. Mathematical & Mechanical Method in Mechanical Engineering Properties of Scalar Product A Frame on a surface

31. Mathematical & Mechanical Method in Mechanical Engineering Frame transformation

32. Mathematical & mechanical Method in Mechanical Engineering Dual Frame, Covariant vectors and Contravariant Vectors Two frames for any vector The two frames are Dual one another

33. Mathematical & mechanical Method in Mechanical Engineering Dual Frame, Covariant vectors and Contravariant Vectors

34. Mathematical & mechanical Method in Mechanical Engineering Dual Frame, Covariant vectors and Contravariant Vectors Other important properties of dual frames

35. Mathematical & mechanical Method in Mechanical Engineering Covariancy and Contravariancy Two frames Transformation T : Coordinate :

36. Mathematical & mechanical Method in Mechanical Engineering Covariancy and Contravariancy Two frames

37. Mathematical & mechanical Method in Mechanical Engineering Covariancy and Contravariancy • Dual frame is changing with the transformation • so it is with the coordinate

38. Mathematical & mechanical Method in Mechanical Engineering Covariancy and Contravariancy

39. Mathematical & mechanical Method in Mechanical Engineering Covariancy and Contravariancy Transformation relationship between Assume

40. Mathematical & mechanical Method in Mechanical Engineering Covariancy and Contravariancy Since

41. Mathematical & mechanical Method in Mechanical Engineering Covariancy and Contravariancy Coordinate Transform

42. Mathematical & mechanical Method in Mechanical Engineering Covariancy and Contravariancy

43. Mathematical & mechanical Method in Mechanical Engineering Covariancy and Contravariancy Overall relationships among frame and coordinate transformations

44. Mathematical & mechanical Method in Mechanical Engineering Covariancy and Contravariancy Components of vector A inare called contra-variant components Components in the dual frame are called a covariant component.

45. Mathematical & mechanical Method in Mechanical Engineering Covariancy and Contravariancy A transformation The Jacobian matrix of transformation is a matrix of contravariance

46. Mathematical & mechanical Method in Mechanical Engineering Covariancy and Contravariancy Example of contravariant quantity is contravariant to the frame transformation

47. Mathematical & mechanical Method in Mechanical Engineering Covariancy and Contravariancy Contravariant quantity under the coordinate transformation

48. Mathematical & mechanical Method in Mechanical Engineering Covariancy and Contravariancy Covariant quantity under the coordinate transformation

49. Mathematical & Mechanical Method in Mechanical Engineering Class is Over! See you!