Dr. Wang Xingbo Fall ， 2005

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

2. Mathematical & Mechanical Method in Mechanical Engineering Differential Forms • Dual space • Wedge Product • “d” operation • 1-form • 2-form • Surface Integral

3. Mathematical & Mechanical Method in Mechanical Engineering Dual space Let V be a real vector space over R; a real linear transformation on V is such a transformation that returns a real-value when it applies on the element v of V, namely

4. Mathematical & Mechanical Method in Mechanical Engineering Dual space We also call such transformation a linear map or linear functional. The dual space of V, denoted by V*, consists of the set of all linear maps from V to R, and satisfies

5. Mathematical & Mechanical Method in Mechanical Engineering Dual space The basis of V is Then the basis of V* is A element in V* is

6. Mathematical & Mechanical Method in Mechanical Engineering Geometric meaning

7. Mathematical & Mechanical Method in Mechanical Engineering Wedge product A wedge product is denoted by symbol ∧, also called exterior product. A wedge product of two quantities  and  is the directed area swept out from  to . Thus the wedge product satisfies the following rules

8. Mathematical & Mechanical Method in Mechanical Engineering “d” Operation Let be two forms and f be a real function. The rules for “d” operation are as follows:

9. Mathematical & Mechanical Method in Mechanical Engineering Differential Form • Complement to vector analysis • many complex formulas and operations in vector analysis can be converted to a simple form by differential forms

10. Mathematical & Mechanical Method in Mechanical Engineering 1-form A differential 1-form A very important example

11. Mathematical & Mechanical Method in Mechanical Engineering 1-form In general, a 1-form on an open set of Rn can be the following form

12. Mathematical & Mechanical Method in Mechanical Engineering 1-form A differential form is very similar to a vector field

13. Mathematical & Mechanical Method in Mechanical Engineering Geometric and Physical Interprets of 1-form A C2 function exists with We will say that it is exact Most differential forms are not exact unless they satisfy If F and G satisfy the above condition, we will call the differential form closed.

14. Mathematical & Mechanical Method in Mechanical Engineering Exactness Exactness is a very important concept. The case occurs frequently in differential equations. Given an equation: If the differential on the left is exact then the curves give solutions to this equation

15. Mathematical & mechanical Method in Mechanical Engineering Related with Line Integrals When is it exact ? It is a closed form on all of R2 with C1 coefficients, then it is exact. Exactness is its path independence

16. Mathematical & mechanical Method in Mechanical Engineering Related with Line Integrals If ω is exact and C1 and C2 are two parameterized curves with the same endpoints (or more accurately the same starting point and ending point), then:

17. Mathematical & mechanical Method in Mechanical Engineering Application in Physics The work done by a force along a displacement If the force and the displacement vary with the position on the path C Thus, if F is a conservative force and C is a close path

18. Mathematical & Mechanical Method in Mechanical Engineering Physical Meanings Given a vector field A function called the potential energy , such that The force is called conservative if it has a potential energy function F is conservative precisely when is exact in terms of differential forms

19. Mathematical & Mechanical Method in Mechanical Engineering Physical Meanings Given a vector field

20. Mathematical & Mechanical Method in Mechanical Engineering 1-form & vectors

21. Mathematical & mechanical Method in Mechanical Engineering 2-forms and Curl of A Vector Field A typical 2-form is as follows

22. Mathematical & mechanical Method in Mechanical Engineering 2-forms and Curl of A Vector Field

23. Mathematical & mechanical Method in Mechanical Engineering “d” of a 1-form and the Curl A C1 1-form is called exact if there is a C2 function f (called a potential) such that then is called closed if If is a closed form on R3 with C1 coefficients, thenωis exact.

24. Mathematical & mechanical Method in Mechanical Engineering 2-forms and Curl of A Vector Field

25. Mathematical & mechanical Method in Mechanical Engineering 2-form and Divergence of a Vector Field converting the above 2-form to the vector field Then the coefficient of

26. Mathematical & mechanical Method in Mechanical Engineering 2-form and Divergence of a Vector Field A 3-form is simply an expression converting the above 2-form to the vector field Then the coefficient of

27. Mathematical & Mechanical Method in Mechanical Engineering 2-form and Divergence of a Vector Field

28. Mathematical & mechanical Method in Mechanical Engineering “d” of a 2-form and Divergence Proposition

29. Mathematical & mechanical Method in Mechanical Engineering Surface Integrals Let S be a smooth parameterized surface

30. Mathematical & mechanical Method in Mechanical Engineering Surface Integrals The integral of a 2-form on S is given by In practice, the integral of a 2-form can be calculated by first converting it to the form and then evaluating

31. Mathematical & Mechanical Method in Mechanical Engineering Forms & Integral Green Theorem Stokes Theorem

32. Mathematical & Mechanical Method in Mechanical Engineering Class is Over See You!