Chapter 9 向量分析 (Vector Analysis) Vector Integration — Line Integrals vector integral scalar integrals only in Cartesian system The integral with respect to x cannot be evaluated unless y and z are known in terms of x and similarly for the integrals with respect to y and z. The path of integration C must be specified, i.e., the integral depends on the particular choice of contour C. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Vector Integration — Line Integrals Example : the force exerted on a body is , Calculate the work done going from the origin to the point (1,1). The integrals cannot be evaluated until we specify the values of y as x and x as y ! (1,1) For this force the work done depends on the choice of path ! (this force is a nonconservative force) (1,0) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Vector Integration — Line Integrals If then 此運算的物理功能並不顯著 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Vector Integration — Surface Integrals and Volume Integrals z The most commonly encountered form n y A flow or flux through the given surface (divergence). Area element x Two conventions for choosing the positive directions : 1. For closed surface, the outward normal is positive. 2. For open surface, obey the right-hand rule. Right-hand rule for the positive normal For the volume element dτ is a scalar quantity, volume integrals are somewhat simpler ! Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Integral Definitions of Gradient, Divergence, and Curl is the volume of a small region of space Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) The proof of the integral definition of gradient : For surface EFHG is outward For surface ABDC z G H C D y E F A Using the first two terms of a Maclaurin expansion B x Differential rectangular parallelepiped (origin at center) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) The proof of the integral definition of divergence : For surface EFHG is outward For surface ABDC z G H C D y y E F A B x Differential rectangular parallelepiped (origin at center) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) The proof of the integral definition of curl : z G H C D y y E F A B x Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Gauss’s Theorem Closed surface Gauss’s theorem states the relation between a surface integral of a function and the volume integral of the divergence of that function. For example : For each parallelepiped Net rate of flow out = terms cancel (pairwise) for all interior faces Only the contributions of the exterior surface survive. Number , dimensions Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Green’s Theorem If u and v are two scalar function, we have the identities : For developing Green’s functions Gauss’s Theorem : Green’s Theorem Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Alternate Forms of Gauss’s Theorem Volume integral involving divergence gradient ? curl ? Suppose is a vector with constant magnitude and constant but arbitrary direction. Volume integral involving gradient (86清華化工,70成大電機) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Alternate Forms of Gauss’s Theorem Volume integral involving curl ? Suppose is a vector with constant magnitude and constant but arbitrary direction. (90成大機械,88交大機械,84清華動機) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Stokes’s Theorem Stokes’s theorem states the relation between a line integral of the function and the (open) surface integral of a curl of that function. y 3 x0, y0+dy x0+dx, y0+dy 4 2 dλ x0, y0 1 x0+dx, y0 x Circulation around a differential loop Exact cancellation on interior paths; no cancellation on the exterior path. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Alternate Forms of Stokes’s Theorem Suppose is a vector with constant magnitude and constant but arbitrary direction. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Alternate Forms of Stokes’s Theorem Suppose is a vector with constant magnitude and constant but arbitrary direction. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Potential Theory – Scalar Potential If a force over a given simply connected region of space S can be expressed as the negative gradient of a scalar function φ then we call φ a scalar potential that describes the force by one function instead of three. The force F appearing as the negative gradient of a single-valued scalar potential is labeled a conservative force. (gravitational and electrical force) Stokes’s theorem for every closed path in our simply connected region S. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Potential Theory – Scalar Potential conservative force D B Physically, this means that the work done in going from A to B is independent of the path and the work done in going around a closed path is zero. A C Energy is conserved ! possible paths for doing work Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Example Gravitational Potential The gravitational force on a unit mass m1 : The potential is the work done in bringing the unit mass in from infinity The final negative sign is a consequence of the attractive force of gravity. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Example Centrifugal Potential The centrifugal force per unit mass : φ φSHO r The simple harmonic oscillation : φG φC Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Thermodynamics – Exact Differentials In thermodynamics depends only on the end points if df is indeed an exact differential The necessary and sufficient condition is that or is irrotational with Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Potential Theory – Vector Potential In electromagnetic theory is a vector potential is solenoidal Suppose Assuming the coordinates have been chosen so that is parallel to the yz-plane, that is Where f2 and f3 are arbitrary functions of y and z but are not functions of x. Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Vector Potential Using the Leibniz formula for the derivative of an integral Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Vector Potential Remembering that f2 and f3 are arbitrary functions of y and z, we choose Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Example A Magnetic Vector Potential For a Constant Magnetic Field Bz is a constant Assuming the coordinates have been chosen so that is parallel to the yz-plane, that is Assuming the coordinates have been chosen so that is parallel to the xy-plane, that is Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) with p any constant is not unique a gauge transformation Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Gauge covariant derivative The Schrödinger equation without magnetic induction field The Schrödinger equation with magnetic induction field Gauge covariant derivative describes the coupling of a charged particle with the magnetic field. Minimal substitution Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Gauss’s Law q Gauss’s theorem s Gauss’s law q s Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Gauss’s Law z S’  spherical s s’ q y x Exclusion of the origin Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Poisson’s Equation Poisson’s equation Laplace’s equation Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Dirac Delta Function Example 1.6.1 Gauss’s theorem Include the origin Not include the origin Dirac delta function : Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) δ-sequence function Lorentzian x x Convenient to differentiate Hermite polynomials Useful in Fourier analysis and quantum mechanics x x Dirichlet kernel Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) δ function From a mathematical point of view, do not exist. The Dirac delta function must be even in x, Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) δ function The definition of the derivative A linear operator £(x0) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Uniqueness Theorem A vector is uniquely specified by giving its divergence and its curl within a simply connected region and its normal component over the boundary. A source (charge) density A circulation (current) density Assuming Let Laplace equation using the Green’s theorem Wn = V1n-V2n = 0 on the boundary Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Helmholtz’s Theorem A vector satisfying , , with both source and circulation densities vanishing at infinity may be written as the sum of the two parts, one of which is irrotational, the other solenoidal. irrotational solenoidal is a known vector We construct a scalar potential and a vector potential If s = 0, then is solenoidal If c = 0, then is irrotational Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Helmholtz’s Theorem z Field point (x1,y1,z1) Source point (x2,y2,z2) If we can show that y x Include the origin Not include the origin Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Helmholtz’s Theorem Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Helmholtz’s Theorem = 0 = 0 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

Chapter 9 向量分析 (Vector Analysis) Helmholtz’s Theorem A vector satisfying , , with both source and circulation densities vanishing at infinity may be written as the sum of the two parts, one of which is irrotational, the other solenoidal. irrotational solenoidal Applied to the electromagnetic field Irrotational electric field Solenoidal magnetic induction field Source density electric charge density divided by electric permittivity circulation density electric current density times magnetic permeability Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung