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Chapter 8 向量分析 ( Vector Analysis)

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.

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Chapter 8 向量分析 ( Vector Analysis)

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  1. 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 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

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

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

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

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

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

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

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

  9. 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 : z Definition : θ y x The scalar product is commutative : Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

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

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

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

  13. 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 Prove it! Magnitude of : Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

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

  15. 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 ! Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

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

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

  18. 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 Fourier space 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 . Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

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

  20. Chapter 7 向量分析 (Vector Analysis) Proof : z = 1 in Let us denote The volume is symmetric in αβ,γ z2 = 1 z = ± 1 For the special case Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

  21. 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 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

  22. Chapter 8 向量分析 (Vector Analysis) A Geometrical Interpretation z Q P φ(x,y,z)= C dr y is perpendicular to x z For a given dφ, is a minimum when it is chosen parallel to (cosθ = 1). Q φ= C2 > C1 For a given , is a maximum when is chosen parallel to . φ= C1 P y is a vector having the direction of the maximum space rate of change of φ. x Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

  23. 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: Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

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

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

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

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

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

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

  30. Chapter 8 向量分析 (Vector Analysis) A Physical Interpretation y 3 x0, y0+dy x0+dx, y0+dy 4 2 1 x0, y0 x0+dx, y0 circulation per unit area x Circulation around a differential loop Vorticity (渦度向量) is labeled irrotational Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

  31. 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: Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

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

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

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

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

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

  37. 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 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

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

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