化學數學（一）

1. 化學數學（一） The Mathematics for Chemists (I) (Fall Term, 2004)(Fall Term, 2005)(Fall Term, 2006)Department of ChemistryNational Sun Yat-sen University

2. Chapter 3 Vector Algebraand Analysis • Definition • Scalar (dot) product • Vector (cross) product • Scalar and vector fields • Applications

3. Content covered in the textbook: Chapter 16 Assignment: pp372-374: 15,17,18,24,25,30,32,35,40,41, 43,47,48,55,58,60

4. a a=AB A Definition (naïve) Vectors are a class of quantities that require both magnitude and direction for their specification. Terminal point Terminal point B Initial point Initial point Unit vector: a vector of unit length. Null vector: a vector of zero length. (its direction is meaningless.)

5. Examples of Vectors • Position, velocity, angular velocity, acceleration • Force, torque, momentum, angular momentum • Electric and magnetic fields, electric and magnetic dipole moments,

6. a b a+b b a a+b b a a-b -b a -0.5a 1.5a -a Vector Algebra Equality: Addition: Subtraction: Scalar multiplication:

7. B D b C C’ a A O Example Show that the diagonals of a parallelogram bisect each other. We need to show that the midpoints of OD and AB coincide.

8. B C b X A O a C’’ C b X B A O a C’ Classroom Exercise Show that the mean of the position vectors of the vertices of a triangle is the position vector of the centroid of the triangle.

9. a θ N P O y a ayj axi j x i Components and Decomposition x=(2,3),y=(4.2,-5.6)

10. azk ayj axi Components and Decomposition(in 3D Space)

11. Vector Algebra Restated Equality: Addition: Subtraction: Scalar multiplication:

12. Example

13. m1 m2 m4 r1 r2 m3 r4 r3 The Center of Mass (Gravity)

14. -q r1 r q r2 Dipole Moments Dependence of reference frame: If the total charge Q is zero (e.g., in a molecule), then Total dipole moment:

15. Electric dipole moments

16. Symmetry and Dipole Moment The total dipole moment of a tetrahedron: r4 r1=(a,a,a), r2=(a,-a,-a), r3=(-a,a,-a), r4=(-a,-a,a) r1 r2 r3

17. azk ayj axi Base Vectors Orthogonal basis: Nonorthogonal basis:

18. Classroom Exercise

19. A Δa B a(t) a(t+Δt) O Scalar Differentiation of a Vector

20. z C r(t) y O b x t a Parametric Representation of a Curve

21. Position, Velocity, Momentum, AccelerationNewton’s Second Law Velocity: Speed: Linear momentum: Kinetic energy: Acceleration: Newton’s second law:

22. Classroom Exercise • Write the expression of momentum in terms of the planar polar coordinates

23. A b θ a B O The Scalar (Dot) Product Proof:

24. Example

25. A b θ a B O b b θ a θ a O O If Classroom exercise

26. Orthogonal and Coincident

27. azk ayj axi Cartesian Base Vectors Orthogonality: Normalization (unit length):

28. Force and Work

29. A F(r) Δr B r(t) F(r+Δr) r+Δr O Force and Work: General Case

30. E μ θ Charges in an Electric Field

31. B m θ Magnetic Moment in a Magnetic Field

32. HOW DO YOU KNOW THEY ARE PARALLEL WITH EACH OTHER?

33. b bsinθ θ a v C=AxB b a A B The Vector (Cross) Product A new vector can be constructed from two given vectors: Its magnitude: Its direction: Right-hand rule:

34. v b -v a Important properties Anti-commutative: Nonassociative: (Proof to be given later) Classroom exercise: If the cross product of two vectors is a zero vector, they must be parallel or antiparallel to each other.

35. azk ayj axi In Cartesian Basis

36. Example

37. Classroom Exercise Calculate the cross product of above two vectors using

38. F θ r A O d Application: Moment of Force (Torque)

39. E q r1 r -q O r2 T E μ An Electric Dipole in an Electric Field

40. B qm r1 r -qm O r2 T B m A Magnetic Dipole in a Magnetic Field

41. v r ω O ω v rsinθ r θ O Angular Velocity In a plane: General case:

42. Exercise Classroom exercise

43. Exercise

46. ω r p θ r A O d Angular Momentum A special case: ωis perpendicular r: (moment of inertia)

47. T B m Conservation of Angular Momentum If T=0, angular momentum is conserved. For nuclear spins: NMR measures how fast a nuclear spin precesses.

48. Scalar and Vector Fields

49. The Gradient of a Scalar Field Vector differential operator The gradient of a scalar field is a vector.

50. f(r+dr) dr f(r) The Meaning of the Gradient Gradient is a convenient vector expression of the derivative of multi-variable functions.