Dr. Wang Xingbo Fall ， 2005

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

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

3. Mathematical & Mechanical Method in Mechanical Engineering Vectors • 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=(x1, x2, x3) has three components x1,x2, x3.

4. Mathematical & Mechanical Method in Mechanical Engineering Algebraic properties Vectors X=(x1, x2, x3), Y=(y1, y2, y3) Scalar multiplication:2X = (2x1, 2x2, 2x3) Addition:X + Y = (x1+ y1, x2+ y2, x3+ y3) The zero vector:0 = (0,0,0) The subtraction:X - Y = (x1- y1,x2- y2,x3- y3)

5. Mathematical & Mechanical Method in Mechanical Engineering Length (magnitude) of a vector Length of X = (x1, x2, x3) is calculated by: A unit vector in the direction of X is:

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

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

8. Mathematical & Mechanical Method in Mechanical Engineering Inner Product A=(a1, a2, a3), B=(b1, b2, b3)

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

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

11. Mathematical & mechanical Method in Mechanical Engineering Geometric Meanings of Cross Product • 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.

12. Mathematical & mechanical Method in Mechanical Engineering Properties ofCross Product • A×B = -B ×A, • A ×(B + C) = A ×B +A ×C, • A||B is the same as A ×B = 0

13. Mathematical & mechanical Method in Mechanical Engineering Three Basis Vectors • i1×i1 = 0, i2×i2 = 0, i3×i3 = 0, • i1×i2 = i3, i2×i3 = i1, i3×i1 = i2 A = a1i1 + a2i2 + a3i3, B = b1i1 + b2i2 + b3i3

14. Mathematical & mechanical Method in Mechanical Engineering Product of Three Vectors • (A×B)×C = B(A·C) -A(B·C) • A×(B×C) = B(A·C) - C(A·B) A = a1i1 + a2i2 + a3i3, B = b1i1 + b2i2 + b3i3, C = c1i1 + c2i2 + c3i3

15. Mathematical & mechanical Method in Mechanical Engineering Other Useful Formula for Vector Products • A·(B×C) = (A×B) ·C = (C×A) ·B • (A× B) · (C×D) = (A·C)(B·D) - (A·D)(B·C) • (A× B) · (A×C)= B·C - (A·C) (A·B) • (A×B) ·(C×D) + (B×C) ·(A×D) + (C×A) ·(B×D) = 0 . • A×(B×C) + B×(C×A) + C×(A×B) = 0 • (A×B) ×(C×D) = C(A·(B×D)) - D(A·(B ×C)) • = B(A·(C×D))-A(B·(C×D))

16. Mathematical & mechanical Method in Mechanical Engineering Vector Calculus • 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 < ∞)

17. Mathematical & mechanical Method in Mechanical Engineering The Derivatives of a Vector Function A(t) = (A1(t),A2(t),A3(t)) = A1(t)i1 + A2(t)i2 + A3(t)i3

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. Mathematical & mechanical Method in Mechanical Engineering The Integral of a Vector Function A(t) = (A1(t),A2(t),A3(t)) = A1(t)i1 + A2(t)i2 + A3(t)i3

21. Mathematical & mechanical Method in Mechanical Engineering Fields • 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 where symbol means “subordinate to”

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

23. Mathematical & mechanical Method in Mechanical Engineering Scalar Fields 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. A scalar field can be intuitionistically described by level surfaces

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

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

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

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

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

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

30. Mathematical & Mechanical Method in Mechanical Engineering Flux A(r)=(A1(x1, x2, x3),A2(x1, x2, x3),A3(x1, x2, x3)) in Cartesian coordinate system

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

32. Mathematical & Mechanical Method in Mechanical Engineering Divergence A(r)=(A1(x1, x2, x3),A2(x1, x2, x3),A3(x1, x2, x3)) In Cartesian coordinate system

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

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

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

36. Mathematical & Mechanical Method in Mechanical Engineering Circulation A(r)=(A1(x1, x2, x3),A2(x1, x2, x3),A3(x1, x2, x3)) l be a orientated close curve

37. Mathematical & Mechanical Method in Mechanical Engineering The Curl of a Vector Field A(r)=(A1(x1, x2, x3),A2(x1, x2, x3),A3(x1, x2, x3))

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

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

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

41. Mathematical & Mechanical Method in Mechanical Engineering Summary

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