1. 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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

36. 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

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

38. 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

39. 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