1. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) 物理量與符號 物理量 : 1. 純量 (scalar quantity): 有大小, 無方向 例如 : 質量 (mass), 溫度 (temperature), 壓力 (pressure), 能量 (energy) 2. 向量 (vector quantity): 有大小以及方向 例如 : 速度 (velocity), 動量 (momentum), 力矩 (torque) 向量符號 : 1. 一般向量 : 長度 2. 單位向量 : 長度 1

2. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) 向量的基本運算 1. 向量之相等 : 包括大小以及方向的相等, 2. 向量之反向 : 大小相等但方向相反, 3. 向量之合成 : 4. 向量之倍數 : 5. 向量之純量積 (scalar product): 功 (work) 的計算 6. 向量之向量積 (vector product): 力矩 (torque) 的計算 向量之合成 : Commutative : Associative :

3. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) 向量的座標表示法 xAxA yAyA zAzA 終點表示法 : 分量表示法 : 單位向量表示法 : x y z

4. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) 座標軸的轉動 (Rotation of the Coordinate Axes) X Y r x y φ X’ Y’ x’ y’ φ

5. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) 座標軸的轉動 (Rotation of the Coordinate Axes) Let The coefficient a ij is the cosine of the angle between x i ’ and x j N dimensions

6. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) 座標軸的轉動 (Rotation of the Coordinate Axes) The coefficient a ij is the cosine of the angle between x i ’ and x j Using the inverse rotation : yieldsor

7. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) The orthogonality condition for the direction cosines a ij : or The Kronecker delta is defined by for j = k for j  k

8. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) Vector and vector space Vector :and 1. Vector equality : means x i = y i, i = 1,2,3. 2. Vector addition : means x i + y i = z i, i = 1,2,3. 3. Scalar multiplication : (with a real). 4. Negative of a vector : 5. Null vector : there exists a null vector

9. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) Scalar (Dot) Product The projection of a vector onto a coordinate axis is a special case of the scalar product of and the coordinate unit vectors : The scalar product is commutative : z x y θ Definition :

10. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) Distributive Law in the Scalar (Dot) Product

11. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) Normal vector is a nonzero vector in the x-y plane x y

12. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) Invariance of the scalar product under rotations (using the indices k and l to sum over x,y, and z) take invariant Scalar quantity

13. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) Vector (Cross) Product Definition : i,j,k all different and with cyclic permutation of the indices i,j, and k Magnitude of : Prove it!

14. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) Parallelogram representation of the vector product x y θ Bsinθ anticommutation

15. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) the vector product under rotations i,j, and k in cyclic order If i = 3, then j = 1, k =2 l  m is indeed a vector !

16. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) Triple Scalar Product The dot and the cross may be interchanged : scalar

17. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) Parallelepiped representation of triple scalar product x y z Volume of parallelepiped defined by,, and

18. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) Construction of a reciprocal crystal lattice Let,, and (not necessarily mutually perpendicular) represent the vectors that define a crystal lattice. The distance from one lattice point to another may be written as With these vectors we may form the reciprocal lattices : We see that is perpendicular to the plane containing and and has a magnitude proportional to. Fourier space

19. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) Triple Vector Product x y z BAC-CAB rule

20. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 7 向量分析 (Vector Analysis) Proof : z = 1 in Let us denote The volume is symmetric in α  β,γ z 2 = 1 z = ± 1 For the special case

21. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) Gradient Suppose that φ(x,y,z) is a scalar point function which is independent of the rotation of the coordinate system. We construct a vector with components : or A vector differential operator

22. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung is a vector having the direction of the maximum space rate of change of φ. Chapter 8 向量分析 (Vector Analysis) A Geometrical Interpretation P Q dr z x y y z x φ(x,y,z)= C φ= C 2 > C 1 φ= C 1 P Q is perpendicular to For a given dφ, is a minimum when it is chosen parallel to (cosθ = 1). For a given, is a maximum when is chosen parallel to.

23. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) Exercise : 試求曲面 上一點 (2,2,8) 之切面與法線方程式 (88 台大造船 ) Solution : 取, 而曲面在 之法向量 為 : 根據直線與平面之點向式 : 切面 : 法線 :

24. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) Example : Calculate the gradient of f(r) =

25. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) Divergence x y Differentiating a vector function : vector : differential property Scalar Vector

26. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) Example : Calculate the divergence of f(r) if

27. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) A Physical Interpretation z x y G C A E B F H D dy dz dx Consider : the velocity of a compressible fluid : the density of a compressible fluid The rate of flow in (EFGH) = The rate of flow out (ABCD) = Expand in a Maclaurin series Net rate of flow out| x = Net rate of flow out = : the net flow of the compressible fluid out of the volume element dxdydz per unit volume per unit time. (divergence) The continuity equation : ρ(x,y,z,t)

28. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) Exercise : For a particle moving in a circular orbit (a)Evaluate (b)Show that (The radius r and the angular velocity are constant) (a) (b) Proof !

29. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) Curl Definition :

30. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) A Physical Interpretation x y x 0 +dx, y 0 x 0 +dx, y 0 +dyx 0, y 0 +dy x 0, y 0 1 2 3 4 Circulation around a differential loop circulation per unit area Vorticity ( 渦度向量 ) is labeled irrotational

31. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) A Physical Interpretation is labeled irrotational (the gravitational and electrostatic forces) Newton’s law of universal gravitation Coulomb’s law of electrostatics Calculate:

32. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) Example : verify that Gradient of a Dot Product BAC-CAB rule

33. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) Successive Applications of The divergence of the gradient : the Laplacian of When φ is the electrostatic potential Laplace’s equation of electrostatics in the European literature 

34. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) Example : Calculate Example : replacing If n = 0 n = -1 A consequence of physics

35. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) Successive Applications of : The curl of the gradient All gradients are irrotational A mathematical identity !

36. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) Successive Applications of : The divergence of a curl All curls are solenoidal

37. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) Successive Applications of : BAC-CAB rule Example : Maxwell’s equation (in vacuum) The electromagnetic vector wave equation

38. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 8 向量分析 (Vector Analysis) Successive Applications of : Exercise : 試證明 (74 台大材料, 清華材料 )