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

2. 1.Dual space 2.Wedge Product 3.“d” operation 4.1-form 5.2-form 6.Surface Integral Mathematical & Mechanical Method in Mechanical Engineering Differential Forms

3. 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 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 Mathematical & Mechanical Method in Mechanical Engineering Dual space

4. 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 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 Mathematical & Mechanical Method in Mechanical Engineering Dual space

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

6. Mathematical & Mechanical Method in Mechanical Engineering Geometric meaning

7. 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 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 Mathematical & Mechanical Method in Mechanical Engineering Wedge product

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

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

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

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

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

13. A C 2 function exists with Mathematical & Mechanical Method in Mechanical Engineering Geometric and Physical Interprets of 1-form 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. Exactness is a very important concept. The case occurs frequently in differential equations. Given an equation: Mathematical & Mechanical Method in Mechanical Engineering Exactness If the differential on the left is exact then the curves give solutions to this equation

15. When is it exact ? Mathematical & mechanical Method in Mechanical Engineering Related with Line Integrals It is a closed form on all of R 2 with C 1 coefficients, then it is exact. Exactness is its path independence

16. If ω is exact and C 1 and C 2 are two parameterized curves with the same endpoints (or more accurately the same starting point and ending point), then: Mathematical & mechanical Method in Mechanical Engineering Related with Line Integrals

17. The work done by a force along a displacement Mathematical & mechanical Method in Mechanical Engineering Application in Physics 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. Given a vector field Mathematical & Mechanical Method in Mechanical Engineering Physical Meanings 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. Given a vector field Mathematical & Mechanical Method in Mechanical Engineering Physical Meanings

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

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

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 C 1 1-form is called exact if there is a C 2 function f (called a potential) such that then  is called closed if If is a closed form on R 3 with C 1 coefficients, then ω is 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. A 3-form is simply an expression 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

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

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

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

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

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

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