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

2. 1.Vectors 2.Products of Two Vectors 3.Vector Calculus 4.Fields 5.Applications of Gradient, Divergence and Curl Mathematical & Mechanical Method in Mechanical Engineering Vector Algebra

3. Quantities that have both magnitude and direction; the magnitude can stretch or shrink, and the direction can reverse. In a 3-dimmensional space, a vector X=(x 1, x 2, x 3 ) has three components x 1,x 2, x 3. Mathematical & Mechanical Method in Mechanical Engineering Vectors

4. Vectors X=(x 1, x 2, x 3 ), Y=(y 1, y 2, y 3 ) Scalar multiplication:2X = (2x 1, 2x 2, 2x 3 ) Addition:X + Y = (x 1 + y 1, x 2 + y 2, x 3 + y 3 ) The zero vector:0 = (0,0,0) The zero vector:0 = (0,0,0) The subtraction: X - Y = (x 1 - y 1,x 2 - y 2,x 3 - y 3 ) Mathematical & Mechanical Method in Mechanical Engineering Algebraic properties

5. Length of X = (x 1, x 2, x 3 ) is calculated by: A unit vector in the direction of X is : Mathematical & Mechanical Method in Mechanical Engineering Length (magnitude) of a vector

6. Proj u A = (|A| cos  )u ( |u| = 1) Mathematical & Mechanical Method in Mechanical Engineering Projection of a Vector

7. 1.Inner Product,doc product,scalar product 2.Vector Product,cross product 3.Without extension Mathematical & Mechanical Method in Mechanical Engineering Products of Two Vectors

8. A=(a 1, a 2, a 3 ), B=(b 1, b 2, b 3 ) Mathematical & Mechanical Method in Mechanical Engineering Inner Product

9. 1.Non-negative law 2. Commutative law: 3.Distributive law: Mathematical & Mechanical Method in Mechanical Engineering Properties of Scalar Product

10. 1.Cross product of two vectors A and B is another vector C that is orthogonal to both A and B 2. C = A × B 3.|C| = |A||B||sin  | Mathematical & Mechanical Method in Mechanical Engineering Vector Product

11. 1.The length of C is the area of the parallelogram spanned by A and B 2. The direction of C is perpendicular to the plane formed by A and B; and the three vectors A, B, and C follow the right-hand rule. Mathematical & mechanical Method in Mechanical Engineering Geometric Meanings of Cross Product

12. 1. A×B = -B ×A, 2. A ×(B + C) = A ×B +A ×C, 3. A||B is the same as A ×B = 0 Mathematical & mechanical Method in Mechanical Engineering Properties of Cross Product

13. 1.i 1 × i 1 = 0, i 2 × i 2 = 0, i 3 × i 3 = 0, 2.i 1 × i 2 = i 3, i 2 × i 3 = i 1, i 3 × i 1 = i 2 Mathematical & mechanical Method in Mechanical Engineering Three Basis Vectors A = a 1 i 1 + a 2 i 2 + a 3 i 3, B = b 1 i 1 + b 2 i 2 + b 3 i 3

14. 1.(A × B) × C = B(A·C) -A(B·C) 2.A × (B × C) = B(A·C) - C(A·B) Mathematical & mechanical Method in Mechanical Engineering Product of Three Vectors A = a 1 i 1 + a 2 i 2 + a 3 i 3, B = b 1 i 1 + b 2 i 2 + b 3 i 3, C = c 1 i 1 + c 2 i 2 + c 3 i 3

15. 1. A · (B×C) = (A×B) · C = (C×A) · B 2. (A× B) · (C×D) = (A · C)(B · D) - (A · D)(B · C) 3. (A× B) · (A×C)= B · C - (A · C) (A · B) 4.(A×B) ·(C×D) + (B×C) ·(A×D) + (C×A) ·(B×D) = 0. 5. A×(B×C) + B×(C×A) + C×(A×B) = 0 6. (A×B) ×(C×D) = C(A · (B×D)) - D(A · (B ×C)) = B(A·(C×D))-A(B·(C×D)) Mathematical & mechanical Method in Mechanical Engineering Other Useful Formula for Vector Products

16. For any scalar t, a function f(t) is called a vector function or a variable vector if there exists a vector corresponding with f(t). A(t) = (cos t, sin t, 0) (- ∞ < t < ∞ ) Mathematical & mechanical Method in Mechanical Engineering Vector Calculus

17. Mathematical & mechanical Method in Mechanical Engineering The Derivatives of a Vector Function A(t) = (A 1 (t),A 2 (t),A 3 (t)) = A 1 (t)i 1 + A 2 (t)i 2 + A 3 (t)i 3

18. Mathematical & mechanical Method in Mechanical Engineering Properties of Vector Derivative velocity acceleration

19. Mathematical & mechanical Method in Mechanical Engineering Properties of Vector Derivative

20. A(t) = (A 1 (t),A 2 (t),A 3 (t)) = A 1 (t)i 1 + A 2 (t)i 2 + A 3 (t)i 3 Mathematical & mechanical Method in Mechanical Engineering The Integral of a Vector Function

21. Suppose Ω be a subspace, P be any point in Ω, if there exists a function u related with a quantity of specific property U at each point P , namely, Ω is said to be a field of U if Mathematical & mechanical Method in Mechanical Engineering Fields where symbol means “subordinate to”

22. 1. Temperature in a volume of material is a temperature field since there is a temperature value at each point of the volume. 2. Water Velocity in a tube forms a velocity field because there is a velocity at each point of water in the tube. 3. Gravity around the earth forms a field of gravity 4. There is a magnetic field around the earth because there is a vector of magnetism at each point inside and outside the earth. Mathematical & mechanical Method in Mechanical Engineering Example of fields

23. A real function of vector r in a domain is called a scalar field. Pressure function p(r) and the temperature function T(r) in a domain D are examples of scalar fields. Mathematical & mechanical Method in Mechanical Engineering Scalar Fields A scalar field can be intuitionistically described by level surfaces

24. Directional Derivative Mathematical & Mechanical Method in Mechanical Engineering Directive derivatives and gradient Directive derivatives and gradient Where l is a unit vector

25. Gradient Mathematical & Mechanical Method in Mechanical Engineering Directive derivatives and gradient Directive derivatives and gradient It can be shown if l is a unit vector

26. Properties 1.The gradient gives the direction for most rapid increase. 2.The gradient is a normal to the level surfaces. 3.Critical points of f are such that  =0 at these points Mathematical & Mechanical Method in Mechanical Engineering Directive derivatives and gradient Directive derivatives and gradient

27. Mathematical & Mechanical Method in Mechanical Engineering Operational rules for gradient Operational rules for gradient

28. Two important concepts about a vector field are flux,divergence, circulation and curl Mathematical & Mechanical Method in Mechanical Engineering Vector Fields Vector Fields A vector field can be intuitionistically described by vector curve tangent at each point to the vector that is produced by the field

29. The Flux is the rate at which some- thing flows through a surface. Mathematical & Mechanical Method in Mechanical Engineering Flux Flux Let A= A (M) be a vector field, S be an orientated surface, A n be normal component of the vector A over the surface S

30. A(r)=(A 1 (x 1, x 2, x 3 ),A 2 (x 1, x 2, x 3 ),A 3 (x 1, x 2, x 3 )) in Cartesian coordinate system Mathematical & Mechanical Method in Mechanical Engineering Flux Flux

31. Rate of flux to volume. In physics called density. Mathematical & Mechanical Method in Mechanical Engineering Divergence Divergence

32. A(r)=(A 1 (x 1, x 2, x 3 ),A 2 (x 1, x 2, x 3 ),A 3 (x 1, x 2, x 3 )) In Cartesian coordinate system Mathematical & Mechanical Method in Mechanical Engineering Divergence Divergence

33. Lagrangian Operator Mathematical & Mechanical Method in Mechanical Engineering Divergence Divergence

34. Mathematical & Mechanical Method in Mechanical Engineering Operational rules for divergence Operational rules for divergence

35. Circulation is the amount of something through a close curve Mathematical & Mechanical Method in Mechanical Engineering Circulation Circulation A(  ) be a vector field, l be a orientated close curve

36. A(r)=(A 1 (x 1, x 2, x 3 ),A 2 (x 1, x 2, x 3 ),A 3 (x 1, x 2, x 3 )) l be a orientated close curve Mathematical & Mechanical Method in Mechanical Engineering Circulation Circulation

37. A(r)=(A 1 (x 1, x 2, x 3 ),A 2 (x 1, x 2, x 3 ),A 3 (x 1, x 2, x 3 )) Mathematical & Mechanical Method in Mechanical Engineering The Curl of a Vector Field The Curl of a Vector Field

38. Makes circulation density maximal at a point along the curl. Makes circulation density maximal at a point along the curl. Mathematical & Mechanical Method in Mechanical Engineering The Curl of a Vector Field The Curl of a Vector Field

39. Mathematical & Mechanical Method in Mechanical Engineering Operational rules for Rotation (Curl) Operational rules for Rotation (Curl)

40. Potential Field Potential Field A=grad  Tube Field divA =0 Harmonic Field divA =0, rotA=0 Mathematical & Mechanical Method in Mechanical Engineering Several Important Fields Several Important Fields

41. Mathematical & Mechanical Method in Mechanical Engineering Summary Summary

42. Class is Over! See you Friday Evening! Mathematical & Mechanical Method in Mechanical Engineering 21:30,1,Dec,2005 21:30,1,Dec,2005