ElectroDynamicsScript.pdf

Courses in Physics Electrodynamics Electricity, Magnetism and Radiation Ph.W. Courteille Universidade de São Paulo Instituto de Fı́sica de São Carlos 06/08/2024 2. 3. 4 Preface The script was developed for the course Electromagnetism A (SFI5708) offered by the Institute of Physics of São Carlos (IFSC) of the University of São Paulo (USP). The course is divided into two parts. Part I of the script gives an introduction to electromagnetism at the level of an undergraduate course. From electrical and mag- netic phenomena observed in experiments we derive the fundamental laws of Gauss, Faraday, Ampère, and Maxwell allowing a complete description of electromagnetism, which culminates in Maxwell’s equations. The procedure in part II of the script is in- verse. From Maxwell’s equations we deduce electromagnetic phenomena, such as the radiation of accelerated charges or the role of electromagnetic forces in atoms. Also, we derive fundamental conservation laws and the relationship of electromagnetism with the theory of special relativity. This part is a postgraduate course in classical electrodynamics. The course is intended for masters and PhD students in physics. The script is a preliminary version continually being subject to corrections and modifications. Error notifications and suggestions for improvement are always welcome. The script incorporates exercises the solutions of which can be obtained from the author. Information and announcements regarding the course will be published on the website: http://www.ifsc.usp.br/ strontium/ − > Teaching − > Semester The student’s assessment will be based on written tests and a seminar on a special topic chosen by the student. In the seminar the student will present the chosen topic in 15 minutes. He will also deliver a 4-page scientific paper in digital form. Possible topics are: - Existence of magnetic monopoles and the quantization of charge, - The Goos-Hänchen and the Imbert-Fedorov shift, - The Abraham-Minkowski dilemma, - The Aharonov-Bohm effect, - Superconductivity and the Meissner effect, - Cerenkov radiation, - Bremsstrahlung, - The Lorentz model of the radiation of an atom (Sec. 7.2.3), - The Drude model for light-metals interaction (Sec. 7.2.5), - The Kramers-Kronig relations (Sec. 7.2.6), - The optical theorem, - Analytical signal, - Quantization of the electromagnetic field, - Optical fibers, - Diffraction through apertures, - Laguerre-Gaussian light modes, - Bessel beams and laser swords, - Free-electron laser, - The ionosphere as a resonant cavity: Schumann resonances, - Excitation of surface plasmon polaritons, - The Faraday effect, 5 - Birefringent crystals and wave plates, - Forbidden photonic bands and photonic crystals, - The Ewald-Oseen theorem, - The Thomas precession, - Anderson localization, - Mie scattering and Mie resonances, - The of coupled dipoles model, - Gaussian optics, - Negative refraction and the perfect lens, - The Kerr effect, - The quantum Hall effect, - Anti-reflective and reflective dielectric coatings, - Hyperbolic metamaterials, - Comparison between electromagnetic waves and matter waves, - Bragg Scattering, - Schlieren photography, - The Fresnel-Fizeau effect, - The Sagnac effect. The following literature is recommended for preparation and further reading: Ph.W. Courteille, script on Optical spectroscopy: A practical course (2020) Ph.W. Courteille, script on Electrodynamics: Electricity, magnetism, and radiation (2020) Ph.W. Courteille, script on Quantum mechanics applied to atomic and molecular physics (2020) J.B. Marion, Classical Eletromagnetic Radiation, Dover (2012) W.K.H. Panofsky and M. Phillips, Classical Electricity and Magnetism, Dover (2012) J.J. Jackson, Classical electrodynamics, John Wiley & Sons (1999) D.J. Griffiths, Introduction to Electrodynamics, Cambridge University Press (2017) J.R. Reitz, F.J. Milford, R.W. Christy, Foundation of electromagnetic theory M. Born, Principles of Optics, 6th ed. Pergamon Press New York (1980) P. Horowitz and W. Hill, The Art of Electronics, Cambridge University Press (2001) U. Tietze & Ch. Schenk, Halbleiterschaltungstechnik, Springer-Verlag (1978) 6 Content 1 Foundations and mathematical formalism 1 1.1 Differential calculus............................ 2 1.1.1 Scalar and vector fields...................... 2 1.1.2 The gradient............................ 2 1.1.3 The divergence........................... 4 1.1.4 The rotation............................ 5 1.1.5 Taylor expansion of scalar and vector fields........... 5 1.1.6 Rules for calculation with derivatives............... 6 1.1.7 Exercises.............................. 7 1.2 Integral calculus.............................. 9 1.2.1 Path integral............................ 9 1.2.2 Surface integral........................... 10 1.2.3 Volume integral........................... 11 1.2.4 Fundamental theorem for gradients................ 11 1.2.5 Stokes’ theorem........................... 12 1.2.6 Gauß’ theorem........................... 13 1.2.7 Exercises.............................. 14 1.3 Curvilinear coordinates.......................... 16 1.3.1 Differential elements in curvilinear coordinates......... 17 1.3.2 Gradient in curvilinear coordinates................ 17 1.3.3 Divergence in curvilinear coordinates............... 18 1.3.4 Rotation in curvilinear coordinates................ 19 1.3.5 Cylindrical coordinates...................... 19 1.3.6 Spherical coordinates........................ 20 1.3.7 Differential operators for tensor fields.............. 21 1.3.8 Exercises.............................. 22 1.4 Differential geometry in curved space................... 26 1.4.1 Co- and contravariant tensors................... 27 1.4.2 Jacobian for coordinate transformations............. 27 1.4.3 Metric and geodesic equation in Euclidean space........ 28 1.4.4 Exercises.............................. 31 1.5 Dirac’s δ-function.............................. 31 1.5.1 The Dirac function in 1 dimension................ 32 1.5.2 The Dirac function in 2 and 3 dimensions............ 33 1.5.3 Analytical signals.......................... 34 1.5.4 Exercises.............................. 37 1.6 Further reading............................... 38 7 8 CONTENT 2 Electrostatics 39 2.1 The electric charge and the Coulomb force................ 39 2.1.1 Quantization and conservation of the charge........... 39 2.1.2 Coulomb’s law........................... 40 2.1.3 Exercises.............................. 41 2.2 Properties of the electric field....................... 45 2.2.1 Field lines and the electric flux.................. 46 2.2.2 Divergence of the electric field and Gauß’ law.......... 47 2.2.3 Rotation of the electric field and Stokes’ law.......... 48 2.2.4 Exercises.............................. 49 2.3 The scalar electrical potential....................... 56 2.3.1 The equations of Laplace and Poisson.............. 57 2.3.2 Potential generated by localized charge distributions...... 58 2.3.3 Electrostatic boundary conditions................ 58 2.3.4 Exercises.............................. 59 2.4 Electrostatic energy............................ 63 2.4.1 Energy of a charge distribution.................. 63 2.4.2 Energy density of an electrostatic field.............. 64 2.4.3 Dielectrics and conductors..................... 65 2.4.4 Induction of charges (influence).................. 66 2.4.5 Electrostatic pressure....................... 67 2.4.6 Exercises.............................. 68 2.5 Treatment of boundary conditions and the uniqueness theorem.... 69 2.5.1 The method of images charges.................. 70 2.5.2 Formal solution of the electrostatic problem........... 71 2.5.3 Green’s Function.......................... 72 2.5.4 Poisson equation with Dirichlet’s boundary conditions..... 73 2.5.5 Poisson equation with von Neumann’s boundary conditions.. 74 2.5.6 Exercises.............................. 74 2.6 Solution of the Laplace equation in situations of high symmetry.... 78 2.6.1 Variable separation in Cartesian coordinates.......... 78 2.6.2 Variable separation in cylindrical coordinates.......... 79 2.6.3 Variable separation in spherical coordinates........... 79 2.6.4 Exercises.............................. 81 2.7 Multipolar expansion............................ 82 2.7.1 The monopole........................... 83 2.7.2 The dipole............................. 84 2.7.3 The quadrupole........................... 84 2.7.4 Expansion into Cartesian coordinates.............. 85 2.7.5 Exercises.............................. 86 2.8 Further reading............................... 91 3 Electrical properties of matter 93 3.1 Polarization of dielectrics......................... 93 3.1.1 Energy of permanent dipoles................... 93 3.1.2 Induction of dipoles in dielectrics................. 95 3.1.3 Macroscopic polarization..................... 96 CONTENT 9 3.1.4 Electrostatic field on a polarized or dielectric medium..... 96 3.1.5 Electric displacement........................ 98 3.1.6 Electrical susceptibility and permittivity............. 99 3.1.7 Exercises.............................. 100 3.2 Influence of charges and capacitance................... 101 3.2.1 Capacitors and storage of electric energy............. 101 3.2.2 Exercises.............................. 103 3.3 Conduction of current and resistance................... 111 3.3.1 Motion of charges in dielectrics and conductors......... 112 3.3.2 Ohm’s law, stationary currents in continuous media...... 112 3.3.3 Exercises.............................. 114 3.4 The electric circuit............................. 115 3.4.1 Kirchhoff’s rules.......................... 116 3.4.2 Measuring instruments....................... 117 3.4.3 Exercises.............................. 117 3.5 Further reading............................... 124 4 Magnetostatics 127 4.1 Electric current and the Lorentz force.................. 127 4.1.1 The Hall effect........................... 129 4.1.2 Biot-Savart’s law.......................... 130 4.1.3 Exercises.............................. 132 4.2 Properties of the magnetic field...................... 136 4.2.1 Field lines and magnetic flux................... 136 4.2.2 Divergence of the magnetic field and Gauß’s law........ 136 4.2.3 Rotation of the magnetic field and Ampère’s law........ 137 4.2.4 Exercises.............................. 139 4.3 The magnetic vector potential....................... 142 4.3.1 The Laplace and Poisson equations................ 142 4.3.2 Magnetostatic boundary conditions................ 144 4.3.3 Exercises.............................. 145 4.4 Multipolar expansion............................ 148 4.4.1 Multipolar magnetic moments................... 148 4.4.2 Exercises.............................. 149 4.5 Further reading............................... 154 5 Magnetic properties of matter 155 5.1 Magnetization................................ 155 5.1.1 Energy of permanent dipoles and paramagnetism........ 155 5.1.2 Impact of magnetic fields on electronic orbits and diamagnetism 156 5.1.3 Macroscopic magnetization.................... 158 5.1.4 Magnetostatic field of a magnetized material.......... 159 5.1.5 The H-field............................. 160 5.1.6 Magnetic susceptibility and permeability............. 161 5.1.7 Exercises.............................. 164 5.2 Induction of currents and inductance................... 167 5.2.1 The electromotive force...................... 167 10 CONTENT 5.2.2 The Faraday-Lenz law....................... 168 5.2.3 Exercises.............................. 170 5.3 Magnetostatic energy............................ 178 5.3.1 Energy density of a magnetostatic field............. 179 5.3.2 Inductors and storage of magnetostatic energy......... 179 5.3.3 Exercises.............................. 180 5.4 Alternating current............................. 183 5.4.1 Electromagnetic oscillations.................... 183 5.4.2 Alternating current circuits.................... 183 5.4.3 Exercises.............................. 185 5.5 Further reading............................... 190 6 Maxwell’s equations 191 6.1 The fundamental laws of electrodynamics................ 192 6.1.1 Helmholtz’s theorem........................ 194 6.1.2 Potentials in electrodynamics................... 194 6.1.3 The macroscopic Maxwell equations............... 195 6.1.4 The fundamental laws in polarizable and magnetizable materials 202 6.1.5 Exercises.............................. 203 6.2 Conservation laws in electromagnetism.................. 206 6.2.1 Charge conservation and continuity equation.......... 207 6.2.2 Energy conservation and Poynting’s theorem.......... 207 6.2.3 Conservation of linear momentum and Maxwell’s stress tensor. 208 6.2.4 Conservation of angular momentum of the electromagnetic field 214 6.2.5 Exercises.............................. 215 6.3 Potential formulation of electrodynamics................. 217 6.3.1 The vector and the scalar potential................ 217 6.3.2 Gauge transformation....................... 218 6.3.3 Green’s function.......................... 221 6.3.4 Retarded potentials of continuous charge distributions..... 222 6.3.5 Retarded fields in electrodynamics and Jefimenko’s equations. 226 6.3.6 The Liénard-Wiechert potentials................. 227 6.3.7 The fields of a moving point charge................ 229 6.3.8 Exercises.............................. 233 6.4 Further reading............................... 235 7 Electromagnetic waves 237 7.1 Wave propagation............................. 237 7.1.1 Helmholtz’s equation........................ 239 7.1.2 The polarization of light...................... 240 7.1.3 The energy density and flow in plane waves........... 243 7.1.4 Slowly varying envelope approximation............. 245 7.1.5 Plane waves in linear dielectrics and the refractive index.... 246 7.1.6 Reflection and transmission by interfaces and Fresnel’s formulas 246 7.1.7 Transfer matrix formalism..................... 252 7.1.8 Exercises.............................. 255 7.2 Optical dispersion in material media................... 261 CONTENT 11 7.2.1 Plane waves in conductive media................. 261 7.2.2 Linear and quadratic dispersion.................. 265 7.2.3 Microscopic dispersion and the Lorentz model.......... 267 7.2.4 Classical theory of radiative forces................ 271 7.2.5 Light interaction with metals and the Drude model....... 277 7.2.6 Causality connecting D ⃗ with E⃗ and the Kramers-Kronig relations279 7.2.7 Exercises.............................. 283 7.3 Plasmons, waveguides and resonant cavities............... 286 7.3.1 Green’s tensor for wave propagation in dielectric media.... 286 7.3.2 Plasmons at metal-dielectric interfaces.............. 291 7.3.3 Negative refraction and metamaterials.............. 294 7.3.4 Wave guides............................. 297 7.3.5 The coaxial line........................... 300 7.3.6 Cavities............................... 300 7.3.7 Exercises.............................. 303 7.4 Beam and wave optics........................... 307 7.4.1 Gaussian optics........................... 307 7.4.2 Non-Gaussian beams........................ 314 7.4.3 Fourier optics............................ 315 7.4.4 Exercises.............................. 319 7.5 Further reading............................... 320 7.5.1 on optics.............................. 320 7.5.2 on metamaterials.......................... 321 8 Radiation 323 8.1 Multipolar expansion of the radiation.................. 323 8.1.1 The radiation of an arbitrary charge distribution........ 323 8.1.2 Multipolar expansion of retarded potentials........... 326 8.1.3 Radiation of an oscillating electric dipole............ 329 8.1.4 Magnetic dipole and electric quadrupole radiation....... 332 8.1.5 Multipolar expansion of the wave equation........... 334 8.1.6 Exercises.............................. 339 8.2 Radiation of point charges......................... 341 8.2.1 Power radiated by an accelerated point charge......... 341 8.2.2 Radiation reaction......................... 345 8.2.3 Exercises.............................. 348 8.3 Diffraction and scattering......................... 352 8.3.1 Coupled dipoles model....................... 352 8.3.2 The limit of the Mie scattering and the role of the refractive index356 8.3.3 Exercises.............................. 358 8.4 Further reading............................... 359 9 Theory of special relativity 361 9.1 Relativistic metric and Lorentz transform................ 362 9.1.1 Ricci’s calculus, Minkowski’s metric, and space-time tensors.. 362 9.1.2 Lorentz transform......................... 364 9.1.3 Contraction of space........................ 366 12 CONTENT 9.1.4 Dilatation of time......................... 366 9.1.5 Transformational behavior of the wave equation......... 367 9.1.6 The Lorentz boost......................... 371 9.1.7 Exercises.............................. 375 9.2 Relativistic mechanics........................... 376 9.2.1 The inherent time of an inertial system............. 376 9.2.2 Adding velocities.......................... 377 9.2.3 Relativistic momentum and rest energy............. 378 9.2.4 Relativistic Doppler effect..................... 380 9.2.5 Relativistic Newton’s law..................... 381 9.2.6 Uniform acceleration........................ 381 9.2.7 Exercises.............................. 382 9.3 Relativistic electrodynamics........................ 384 9.3.1 Relativistic current and magnetism................ 384 9.3.2 Electromagnetic potential and tensor............... 387 9.3.3 Lorentz transformation of electromagnetic fields........ 388 9.3.4 Energy and momentum tensor.................. 390 9.3.5 Solution of the covariant wave equation............. 391 9.3.6 Emission of radiation by a charged particle in gravity..... 392 9.3.7 Exercises.............................. 393 9.4 Lagrangian formulation of electrodynamics............... 393 9.4.1 Relation with quantum mechanics................ 393 9.4.2 Classical mechanics of a point particle in a field......... 394 9.4.3 Generalization to relativistic mechanics............. 397 9.4.4 Symmetries and conservation laws................ 399 9.4.5 Exercises.............................. 400 9.5 Relativistic gravity............................. 401 9.5.1 Metric and geodesic equation in curved space-time....... 402 9.5.2 Schwarzschild metric........................ 403 9.5.3 Christoffel symbols for relativistic space-time, geodesic equation 403 9.5.4 Exercises.............................. 404 9.6 Further reading............................... 404 10 Appendices to ’Electrodynamics’ 405 10.1 Special topic: Goos-Hänchen shift with light and matter waves.... 405 10.1.1 Evanescent wave potentials.................... 405 10.1.2 Energy flux in the evanescent wave................ 406 10.1.3 Imbert-Fedorov shift........................ 408 10.1.4 Matter wave Goos-Hänchen shift at a potential step...... 408 10.2 Special topic: The dilemma of Abraham and Minkowski........ 409 10.2.1 Calculation of the momentum of light in a dielectric medium. 410 10.2.2 Exercises.............................. 411 10.3 Special topic: Advanced Gaussian optics................. 411 10.3.1 Laguerre-Gaussian beams..................... 411 10.3.2 Exercises.............................. 416 10.4 Special topic: Superconductivity..................... 417 10.4.1 London model of superconductivity and the Meissner effect.. 417 0 CONTENT 10.4.2 BCS theory............................. 419 10.4.3 Josephson junctions........................ 421 10.4.4 Synchronization of coupled Josephson junctions......... 422 10.4.5 Exercises.............................. 425 10.5 Quantities and formulas in electromagnetism.............. 427 10.5.1 Electromagnetic quantities.................... 427 10.5.2 Formulas of special relativity................... 429 10.5.3 CGS units.............................. 429 10.6 Rules of vector analysis.......................... 430 10.6.1 Basic rules............................. 430 10.6.2 Deduced rules............................ 431 10.6.3 Integral rules............................ 432 10.7 Rules for Laplace and Fourier transforms................ 433 10.7.1 Laplace transform......................... 433 10.7.2 Correlation............................. 433 10.7.3 Fourier transform.......................... 434 10.7.4 Convolution............................. 436 10.7.5 Green’s functions.......................... 437 10.8 Further reading............................... 438 10.8.1 on the Abraham-Minkowski dilemma............... 438 Chapter 1 Foundations and mathematical formalism The electrodynamic force is one of the four fundamental forces, together with gravi- tation, the strong nuclear force, and the weak nuclear force. It is a long-range force (F ∝ r−2 ) in the same way as gravitation, but unlike nuclear forces, which are short- ranged. Unlike gravitation, it can be attractive or repulsive. The experimentally Figure 1.1: The four known fundamental forces. observed fact, that two spatially separated bodies can exert mutual forces (beyond gravitation), is not explained within classical mechanics. It is necessary to introduce a new degree of freedom called electric charge which, to take account of the existence of attractive and repulsive forces, must exist in two different types called positive or neg- ative charges. Identical charges repel each other, different charges attract each other. Other observations suggest that the charge is a conserved and quantized quantity. Electrodynamics is an field theory, that is, it can describe all electric or magnetic phenomena observed in the following way: Every charge gives rise to a force field, ⃗ which accelerates other charges. But other experimental ob- called field electric E, servations suggest the existence of another force field, called the magnetic field B, ⃗ whose existence is necessary to understand forces only acting on moving charges. That is, the electric and magnetic fields are introduced to explain the forces named after Coulomb and Lorentz, F = q E⃗ + v × B⃗. (1.1) Thus, fields are quantities distributed in space, which in addition can vary in time, F = F(r, t). (1.2) 1 2 CHAPTER 1. FOUNDATIONS AND MATHEMATICAL FORMALISM The concept of a field represents a powerful mathematical tool for describing forces, which are the only observable magnitudes of electromagnetism. That is, we have no sense to see the electricity. We can only infer their existence from the observa- tion of forces. On the other hand, the formulation of electrodynamics via vectorial force fields, can be replaced by a description via potentials, which are either scalar or vectorial fields. In many circumstances, potentials facilitate the resolution of elec- trodynamic problems, but it is important to keep in mind, that potentials are not directly observable. Maxwell’s electrodynamics has a very deep relationship to Einstein’s theory of special relativity, such that each theory is conditioned to the validity of the other. The relativistic formulation allows to distill the symmetry inherent to electrodynamics in a highly aesthetic way. In view of the fundamental role played by scalar and vector fields in electrody- namics, we will start this course the basic mathematical notions of field theory, that is, differential and integral calculus with fields in Cartesian or curvilinear coordi- nates. We will also have to review basic notions of complex numbers and the Dirac distribution. 1.1 Differential calculus 1.1.1 Scalar and vector fields The most basic application of vectors is the designation of positions in space, r = xêx + yêy + zêz. But other physical quantities may also depend on the position where they are measured. In case the quantity varying with position is a scalar, Φ = Φ(r), we speak of scalar field. An example for a scalar field is the temperature distribution across a room. In the case the quantity is a vector, A = A(r), we speak of vector field. Light propagating through space is an example for a vector field. A position is generally defined with respect to the center of the coordinate system, called the origin, such that the distance from the center is given by, √ p r ≡ r · r = x2 + y 2 + z 2 , (1.3) with êr being a unit vector pointing in the direction of r. In electrodynamics we will often deal with quantities (fields) that depend on the distance between a source located at a position r′ and a detector placed at a position r, such as Φ(R) = Φ(r−r′ ), (x − x′ )êx + (y − y ′ )êy + (z − z ′ )êz êR = p. (1.4) (x − x′ )2 + (y − y ′ )2 + (z − z ′ )2 1.1.2 The gradient The derivative of a one-dimensional function Φ(x) measures, how fast the function changes when we move the position x. That is, when we change x by an amount dx, Φ changes by an amount dΦ given by, dΦ dΦ = dx. (1.5) dx 1.1. DIFFERENTIAL CALCULUS 3 Of course it gets trickier, when Φ is a field depending on three coordinates, because we need to specify in which direction we are changing the position. We have, ∂Φ ∂Φ ∂Φ dΦ = dx + dy + dz. (1.6) ∂x ∂y ∂z This equation resembles the scalar product because, ∂Φ ∂Φ ∂Φ dΦ = êx + êy + êz · (dxêx + dyêy + dzêz ) ≡ ∇Φ · dr , (1.7) ∂x ∂y ∂z where we defined a new operator called nabla,   ∂/∂x ∇ ≡ ∂/∂y . (1.8) ∂/∂z The three-dimensional derivative ∇Φ is called the gradient of the scalar field Φ, ∂Φ ∂Φ ∂Φ ∇Φ(r) = êx + êy + êz , (1.9) ∂x ∂y ∂z and it measures the variation of the value of the field from Φ to Φ + dΦ, when we move the vector by an infinitesimal amount between two points r and r + dr. We understand the geometric interpretation of the gradient through its formula- tion as a scalar product: dΦ = ∇Φ · dr = |∇Φ| · |dr| cos θ , (1.10) where θ is the angle between the gradient and the infinitesimal displacement. Now, we fix a magnitude of the displacement |dr| and look for the direction θ in which the variation dΦ is maximum. Obviously, we find the direction θ = 0, that is, when the gradient points in the same direction as the predefined displacement. The gradient of a scalar field Φ(r) calculated at a point r indicates the direction of the greatest field variation from this point, and its absolute value is a measure for the variation. The concept of the gradient is easy to understand in a two-dimensional landscape: Imagine being on the slope of a mountain. Depending on the direction in which you are heading and the duration of the journey dr, you will gain or lose a certain amount of potential energy dΦ, which you can calculate by the scalar product ∇Φ · dr. If the direction chosen is that indicated by the gradient, you will lose (or gain) a maximum of potential energy. If you choose to go in a direction perpendicular to the gradient, that is, along an equipotential line, the potential energy remains unchanged. This is illustrated in Fig. 1.2. Let us consider the example of a parabolic field, Φ(r) = −r2 :   −2x ∇(−r2 ) = −2y  = −2r. (1.11) −2z 4 CHAPTER 1. FOUNDATIONS AND MATHEMATICAL FORMALISM Figure 1.2: The gradient indicates the direction of the largest field variation Φ. We find that at all points of space the variation is faster in radial direction. Although the operator ∇ has the shape of a vector, it has no meaning by itself. In fact, it is a vector operator, that is, a mathematical prescription telling us what to do with the scalar field on which it acts. Nevertheless, it assimilates all the properties of a vector. (We will see in quantum mechanics, that this is more than a coincidence.) Thus, in a way similar as done for the gradient of scalar fields, we can try to apply the ∇ operator on vector fields using the definitions of the scalar and vector products, grad Φ(r) ≡ ∇Φ(r) and div A(r) ≡ ∇ · A(r) rot A(r) ≡ ∇ × A(r). and (1.12) We practice the calculation with the ∇ operator in the Excs. 1.1.7.1 to 1.1.7.3. 1.1.3 The divergence Let us now analyze the possible meaning of the expression ∇ · A called divergence. It is easy to show, ∂Ax ∂Ay ∂Az ∇ · A(r) = + +. (1.13) ∂x ∂y ∂z Obviously the divergence is a scalar field calculated from a vector field. The divergence measures how much a vector field A(r) spreads out starting from a point r. For a given infinitesimal volume it measures the difference between the number of incoming and outgoing field lines. Exposed to a field with divergence, an extended distribution of masses will start to concentrate (spread out) in case of a drain (source). The field lines trace the masses trajectories. Example 1 (Divergence of a radial field ): We consider the example of the radial field, A(r) = r: ∂x ∂y ∂z ∇·r= + + =3. (1.14) ∂x ∂y ∂z 1.1. DIFFERENTIAL CALCULUS 5 1.1.4 The rotation Let us now examine the possible meaning of the expression ∇ × A called rotation. It is easy to show, êx êy êz ∂Az ∂Ay ∂Ax ∂Az ∂Ay ∂Ax ∇×A(r) = ∂x ∂y ∂z = êx − +êy − +êz −. ∂y ∂z ∂z ∂x ∂x ∂y Ax Ay Az (1.15) Obviously the rotation is a vector field calculated from another vector field. The rotation measures how many of the field lines of a vector field A(r) passing through an infinitesimal volume, return into it. Exposed to a field with rotation, an extended distribution of masses will start spinning in closed orbits. We consider the examples shown in Fig. 1.3. The properties of divergence and rotation are complementary. There are fields exhibiting only one of the properties, or both, or none of them. In cases where there is rotation, it is problematic to specify equipotential lines: Either, the field lines are not orthogonal to the equipotential lines, or they come back. (a) (b) (c) (d) (e) Figure 1.3: (a) Field without divergence, (b) with constant divergence, (c) with radial divergence, (d) with rotation, and (e) with rotation. Example 2 (Rotation of a radial field ): We consider the example of a radial field, A(r) = −yêx + xêy :   0 − ∂z x ∇ × A =  ∂z (−y) − 0  = 2êz. (1.16)   ∂x x − ∂y (−y) We practice the calculation with divergence and rotation in the Excs. 1.1.7.4 to 1.1.7.7. 1.1.5 Taylor expansion of scalar and vector fields We know well the Taylor expansion of functions of one variable: ∞ X d Φ(x+h) = exp(h dx )Φ(x) = 1 d ν ν! (h dx ) Φ(x) = Φ(x)+hΦ′ (x)+ h2 Φ′′ (x)+.... (1.17) ν=0 6 CHAPTER 1. FOUNDATIONS AND MATHEMATICAL FORMALISM The generalization of the expansion to a scalar field, which depends on a vector, is, X∞ 1 Φ(r + h) = exp(h · ∇r )Φ(r) = (h · ∇r )ν Φ(r) (1.18) ν=0 ν! = Φ(r) + (h · ∇r )Φ(r) + 21 (h · ∇r )(h · ∇r )Φ(r) +.... We see that the operator ∇r generates a translation. We study the Taylor expansion of scalar fields in Exc. 1.1.7.8. The generalization of the gradient of a vector field is the Jacobian,    ∂A1  A1 ∂x1 · · · ∂A ∂xn 1   .. . A = ...  =⇒ J[A] = .......  (1.19) ∂An ∂An An ∂x1 · · · ∂xn Therefore, the generalization of the expansion to a vector field is,   (h · ∇r )A1 ..  A(r + h) = exp(h · ∇r )A(r) = A(r) + .  +... (1.20) (h · ∇r )An  ∂F1 ∂F1  h1 ∂x1 +... + hn ∂xn ..  = A(r) + .  +... = A(r) + J[A]h +.... h1 ∂F ∂Fn ∂x1 +... + hn ∂xn n 1.1.6 Rules for calculation with derivatives In total there are four possible ways of defining products involving scalar and vector fields, ΦΨ, ΦA, A · B, and A × B, and six product rules to calculate the following expressions, ∇(ΦΨ) , ∇(A·B) , ∇·(ΦA) , ∇·(A×B) , ∇×(ΦA) , ∇×(A×B). (1.21) Second derivatives can also be defined in six different combinations, ∇ · (∇ϕ) , ∇ × (∇ϕ) , ∇(∇ · A) , ∇ · (∇ × A) , ∇ × (∇ × A). (1.22) As these rules are used frequently, we summarized them in Secs. 10.6.1 and 10.6.2. The rules can be derived componentwise from scalar product rules. Very useful tools for this are the Kronecker symbol and the Levi-Civita tensor. Let us consider a Cartesian coordinate system i = 1, 2, 3. The coordinates in this system are xi and ∂ the derivatives ∂i ≡ ∂x i. The Kronecker symbol is defined by, ( 1 for m = n δmn =. (1.23) 0 else The Levi-Civita tensor is defined by,   1 when (kmn) is an even permutation of (123) ϵkmn = −1 when (kmn) is an odd permutation of (123). (1.24)   0 when at least two indices are identical 1.1. DIFFERENTIAL CALCULUS 7 Adopting Einstein’s summing convention, we automatically take the sum of an ex- pression over all indexes appearing twice. For example, the scalar product can be written, X A·B= Ai B i ≡ Ai B i. (1.25) i For the vector product we obtain, (A × B)k ≡ ϵkmn Am Bn. (1.26) Other examples will be discussed in the Excs. 1.1.7.9 to 1.1.7.11. 1.1.7 Exercises 1.1.7.1 Ex: Differential operators Find the gradients of the following scalar fields: a. Φ(r) = x2 + y 3 + z 4 , b. Φ(r) = x2 y 3 z 4 , c. Φ(r) = ex sin y ln z. 1.1.7.2 Ex: 2D landscape A 2D landscape is parametrized by h(x, y) = 10(2xy − 3x2 − 4y 2 − 18x + 28y + 12). a. Where is mountain top? b. What is its height? 1.1.7.3 Ex: Differential operators Calculate ∇r′ |r − r′ |n. 1.1.7.4 Ex: Differential operators 2 Calculate the divergence and the rotation of the vector field A = e−x y êx + 1+y z 2 êy + xêz at the position (0, 1, 1). 1.1.7.5 Ex: Sources and vertices a. Determine the divergence and the rotation of the vector field A = Ax êx + Ay êy + Az êz. b. Calculate for the following fields the sources and vortices: A1 = −yêx + xêy , A2 = +yêx + xêy , A3 = +xêx + yêy , A4 = +xêx + xêy. c. Make a graphic illustration of the fields and give a geometric interpretation of div and rot. 1.1.7.6 Ex: Sources and vertices r Calculate the divergence ∇ · r3. 8 CHAPTER 1. FOUNDATIONS AND MATHEMATICAL FORMALISM 1.1.7.7 Ex: Chain rule for functions of vector field Apply the chain rule to the√gradient of a scalar function of a vector field: ∇ϕ(E(r)). Use the rule to calculate ∇ ar2. 1.1.7.8 Ex: Taylor expansion in 3D Consider the function, 1 f (x) =. |d − x| Calculate the Taylor expansion in x of this function in Cartesian coordinates at the position x = 0 (in all three spatial coordinates) up to second-order. 1.1.7.9 Ex: Levi-Civita tensor Prove the following relationships for the Kronecker symbol and the Levi-Civita tensor by distinguishing the cases in the indices, a. ϵijk δij = 0 , b. ϵijk ϵijk = 6 , d. ϵijk ϵimn = δjm δkn − δjn δkm , c. ϵijk ϵijn = 2δkn. 1.1.7.10 Ex: Levi-Civita tensor Let the vectors A, B, C, and D ∈ R3 be given. Using the Kronecker Symbol and the Levi-Civita Tensor a. show {A × B}i = ϵijk Aj Bk ; b. prove the relationship, (A × B) · C = (B × C) · A = (C × A) · B; c. Using the formulas of (b) derive the following rules of calculation: i. (A × B)2 = A2 B2 − (A · B)2 ii. (A × B) · (C × D) = (A · C)(B · D) − (A · D)(B · C) ; d. prove that: 2 i. (A × B) · [(B × C) × (C × A)] = [A · (B × C)] ii. A × (B × C) + B × (C × A) + C × (A × B) = 0. 1.1.7.11 Ex: Levi-Civita tensor and vector tautologies Be Ψ and Φ scalar fields and A, B, C, and D vector fields. Show the following identities with the help of the Kronecker symbol.: a. A · (B × C) = B · (C × A), b. (A × B) · (C × D) = (A · C)(B · D) − (B · C)(A · D), c. (A × B) × (C × D) = ((A × B) · D)C − ((A × B) · C)D, d. ∇(ΦΨ) = Φ∇Ψ + Ψ∇Φ, 1.2. INTEGRAL CALCULUS 9 e. ∇ × (ΦA) = (∇Φ) × A + Φ∇ × A, f. ∇ × (A × B) = (B · ∇)A − (A · ∇)B + A(∇ · B) − B(∇ · A), g. ∇(A · B) = A × (∇ × B) + B × (∇ × A) + (A · ∇)B + (B · ∇)A, h. ∇ · (∇Φ) = ∆Φ, i. A · (∇Φ) = (A · ∇)Φ, j. A × (∇Φ) = (A × ∇)Φ. k. ∇ · (A × B) = B · (∇ × A) − A · (∇ × B), l. ∇ (ΨA) = A · ∇Ψ + Ψ∇ · A, m. ∇ · (Ψ∇Ψ) = Ψ∆Ψ + (∇Ψ)2 n. ∇ · (A × B) = B · (∇ × A) − A · (∇ × B), o. ∇ · (∇ × A) = 0, p. ∇ × (∇Φ) = 0, q. ∇ × (∇ × A) = ∇(∇ · A) − ∇2 A. 1.2 Integral calculus Three types of integrals are often used in electrodynamics, the path integral, the surface integral, and the volume integral. 1.2.1 Path integral The path integral is defined on a trajectory C(a, b) through a (scalar or vector) field linking a start point a to an end point b. While following the path point by point, incrementing the infinitesimal displacement vector dl (see Fig. 1.4), we evaluate the local value and the direction of the field, multiply it with dl, and sum it up, Z Z Φdl , A · dl. (1.27) C(a,b) C(a,b) Note that in case of a vector field, the integral is taken over the scalar product between theR local field vector and the path element. The work exerted by a force field, W ≡ F · l is an example. For a path through a field crossing all force lines under right angle, the path integral zeroes, meaning that no work is accumulated. Depending on the properties of the field, the integral may only depend on the points a and b and not on the path C chosen to go from one to the other. In this case, we say that the vector field is conservative, but this is not always the case. Choosing a = b we get a closed path, which can be thought of as delimiting a surface in 3D space. We use the notation, I A · dl , (1.28) ∂S where the symbol ∂S suggests, that the path goes along the edge of the surface S. In practice, it is often useful to find a parametrization l(t) for the path with a parameter (e.g. time) defined over t ∈ [0, 1]. It allows us to calculate explicitly, Z Z 1 dl(t) ∇Φ · dl = ∇Φ(r) · dt. (1.29) C(a,b) 0 dt 10 CHAPTER 1. FOUNDATIONS AND MATHEMATICAL FORMALISM Figure 1.4: Integrating along a path in three-dimensional space. Example 3 (Path integral ): As an example, we will calculate the integral along the path parametrized by l(t) = êx cos t + êy sin t, which is a unit circle around the origin, inside the field A(r) = −yêx + xêy : I Z 2π dl A · dl = (−yêx + xêy ) · dt (1.30) 0 dt Z 2π Z 2π d cos t d sin t = (−êx sin t + êy cos t) êx + êy dt = (sin2 t + cos2 t)dt = 2π. 0 dt dt 0 We calculate other examples of path integrals in the Excs. 1.2.7.1 to 1.2.7.4. 1.2.2 Surface integral The surface integral is defined on a surface S, which can be folded in three-dimensional space. The surface is parceled into infinitesimal areas dS, the local value and the direction of the field are evaluated, multiplied with dS, and summed up, Z Z ΦdS , A · dS. (1.31) S S The vector of the area dS is the local normal vector. In case of the vector field, the integral is taken over the scalar product between R the field and the local area. The flux, that is the field lines crossing a surface, Ψ ≡ E · dS it is an example. The normal vector of the surface of a volume is usually taken as pointing out of the volume. For example, a surface element of the x-y plane can be written dS = êz dxdy in Cartesian coordinates. For curved surfaces or in curvilinear coordinates the expression will be more complicated. Often we consider closed surfaces, which can be considered as delimiting a volume in 3D space. We use the notation, I A · dS , (1.32) ∂V where the symbol ∂V suggests that the surface encloses the volume V. Example 4 (Flow of a field ): As an example, we calculate the field flux 1.2. INTEGRAL CALCULUS 11 A = −yêx + x2 yêy through the unit cube: I Z 1 Z 1 Z 1 Z 1 Z 1 Z 1 A · dS = A|z=1 êz dxdy + A|z=−1 (−êz )dxdy + A|x=1 êx dydz cubo −1 −1 −1 −1 −1 −1 Z 1 Z 1 Z 1 Z 1 Z 1 Z 1 + A|x=−1 dydz + A|y=1 êy dzdx + A|y=−1 (−êy )dzdx −1 −1 −1 −1 −1 −1 Z 1 Z 1 Z 1 Z 1 Z 1 Z 1 Z 1 Z 1 =0+0+ (−y)dydz + ydydz + x2 dzdx + x2 dzdx −1 −1 −1 −1 −1 −1 −1 −1 8 =. (1.33) 3 We calculate other examples of surface integrals in Excs. 1.2.7.5 to 1.2.7.8. 1.2.3 Volume integral The volume integral defined by, Z Z ΦdV , AdV. (1.34) V V R R R In the case of a vector field, we simply write: êx Ax dV + êy Ay dV + êz Az dV. Example 5 (Integral de volume): As an example, we calculate the mass of a cube with homogeneous density ρ0 : Z a/2 Z a/2 Z a/2 m= ρ0 dxdydz = a3 ρ0. (1.35) −a/2 −a/2 −a/2 We calculate another example of a volume integral in Exc. 1.2.7.9. 1.2.4 Fundamental theorem for gradients The fundamental theorem of infinitesimal calculus says, Z fb Z b df = F (x)dx = f (b) − f (a) or df = F (x)dx , (1.36) fa a df for F (x) = dx. That is, derivation and integration are inverse operations. Now in vector analysis, as explained above, we know three different types of deriva- tives. For each one we need to formulate the fundamental theorem in a specific way. For gradients, Z ∇Φ · dl = Φ(b) − Φ(a) or dΦ = ∇Φ · dl. (1.37) C(a,b) Since the right-hand side does not depend on the path C, the integral of the gradient can not either. As a consequence, I ∇Φ · dl = 0. (1.38) C 12 CHAPTER 1. FOUNDATIONS AND MATHEMATICAL FORMALISM The geometric interpretation of the fundamental theorem for gradients is simple: Climbing a mountain following a path step by step and gaining at each step the potential energy dΦ = ∇Φdx, we accumulate between the end and the start point of the path the energy Φ(b) − Φ(a). Path independence is an inherent property of gradients. Example 6 (Fundamental theorem for gradients): Let us consider the following example. To travel inside the potential Φ(r) = xy 2 between the points r1 = (0, 0, 0) and r2 = (2, 1, 0), we can choose between several paths, f.ex. l1 (t) = êx 2t + êy t or l2 (t) = êx 2t + êy t2 with t ∈ [0, 1]. In both cases we gain the same potential energy Φ(r2 ) − Φ(r1 ) = 2: Z Z 1 Z 1 ∇Φ · dl1 = (êx y 2 + êy 2xy) · (êx 2 + êy )dt = (2t2 + 4t2 )dt = 2 C(r1 ,r2 ) 0 0 (1.39) Z Z 1 Z 1 ∇Φ · dl2 = (êx y 2 + êy 2xy) · (êx 2 + êy 2t)dt = (2t4 + 8t4 )dt = 2. C(r1 ,r2 ) 0 0 An example for the application of the fundamental theorem for gradients is discussed in Exc. 1.2.7.10. 1.2.5 Stokes’ theorem Stokes’ theorem allows us to convert a surface integral into a path integral provided the field to be integrated can be expressed as a rotation, Z I (∇ × A) · dS = A · dl. (1.40) S ∂S To find a geometric interpretation we remember that the rotation measures the twist of a field A. The integral over the rotation within a given surface (or, more precisely, the flux of the rotation through this surface) measures the total amount of vorticity. A rotating region is like a kitchen beater stirring the surface of an incom- pressible liquid: The more beaters are in the area, the more the liquid will be moved along the edges of the area. Instead of measuring the number of beaters (left-hand side of the theorem (1.40)), we can also walk along the edge of the area and measure the flux along the rim (right-hand side of the theorem (1.40)). An interesting consequence of Stokes’ theorem is that the path integral is inde- pendent of the shape of the surface. That is, if the field to be integrated can be expressed bin terms of a rotation, we can deform the surface (without touching the edge) without changing the twist of the field, I (∇ × A) · dS = 0. (1.41) S This is analogous to the corollary obtained for gradients (1.38). Example 7 (Teorema de Stokes): Consider the following example. A field be given by, A = −yêx + xêy , such that ∇ × A = 2êz. The surface be a disk of 1.2. INTEGRAL CALCULUS 13   R cos ωt radius R enclosed by a circular path parametrized by, l =  R sin ωt . Then,   0     I Z 2π Z 2π/ω −y −Rω sin ωt ˙ = 2 A · dl = A · ldt  x  ·  Rω cos ωt  dt = 2πR     circle 0 0 0 0   Z Z 0 Z (∇ × A) · dS = 0 êz dA = 2 dA = 2πR2.   disk disk disk 2 In the Excs. 1.2.7.11 and 1.2.7.12 we show applications of Stokes’ theorem. 1.2.6 Gauß’ theorem Gauß theorem allows us to convert a volume integral into a surface integral provided the field to be integrated can be expressed by a divergence, Z I (∇ · A) · dV = A · dS. (1.42) V ∂V To find a geometric interpretation we remember that the divergence measures the expansion force of the field A. The integral over the divergence within a given volume measures the total amount of expansion. A divergent region with is like a tap releasing an incompressible liquid: The more taps are in the volume, the more liquid will be expelled by the edges of the volume. Instead of measuring the number of taps (left-hand side of the theorem (1.42)), we can also bypass the volume by measuring the flux through the surface (right-hand side of the theorem (1.42)). Figure 1.5: Illustration of the theorems of Gauß (above) and Stokes (below). Example 8 (Gauß theorem): Consider the following example. A field be given by, A = r, such that ∇ · A = 3. The volume be a sphere of radius R 14 CHAPTER 1. FOUNDATIONS AND MATHEMATICAL FORMALISM enclosed by a surface. So: I Z Z 2π Z π A · dS = r · êr dS = rr2 sin θdθdϕ = 4πR3 sphericalsurf ace sphericalsurf ace 0 0 Z Z 4π (∇ · A)dV = 3dV = 3 R3 = 4πR3. sphere sphere 3 In the Excs. 1.2.7.13 to 1.2.7.16 we show applications of Gauß’ theorem. The theorems of Stokes and Gauß are often used in the context of cylindrical or spherical coordinates. Therefore, we will postpone the presentation of further examples, until we have discussed curvilinear coordinates. 1.2.7 Exercises 1.2.7.1 Ex: Path integral and work Be the field electric A(r) = E0 zêz be given. A charge +q be shifted on a straight line from the point (0, 0, 0) to the point (1, 1, 1). a. Write down parametrization for the trajectory. b.R Calculate the work spent on this charge explicitly along the path integral W = q E(r) · dr. c. Calculate the work via the potential ϕ. 1.2.7.2 Ex: Path integral and work Consider a field E⃗ depending on z in the following way E⃗ = E0 zêz. A charge q is moved on a spiral-shaped trajectory r(t) with radius R,   R cos t r(t) =  R sin t  h 6π t between z = 0 until z = h. Make a scheme of r(t). Calculate work made on the R charge explicitly via a path integral W = q E⃗ · dr. How can we calculate work more easily? 1.2.7.3 Ex: Path integral and work Calculate the path integral in the field Φ = x2 êx + 2yzêy + y 2 êz from the origin to the point (1, 1, 1) on three different paths: a. For the path (0, 0, 0) −→ (1, 0, 0) −→ (1, 1, 0) −→ (1, 1, 1); b. for the path (0, 0, 0) −→ (0, 0, 1) −→ (0, 1, 1) −→ (1, 1, 1); c. and on a straight line. 1.2.7.4 Ex: Curve parametrization The movement of a mass point is given in Cartesian coordinates by the vector r(t) = (ρ cos ϕ(t), ρ sin ϕ(t), z0 ) with ρ = vt and ϕ = ωt + ϕ0. What is the geo- metric figure dashed by the movement? Express the speed ṙ(t) and the acceleration r̈(t) in Cartesian coordinates. Calculate |r(t)|2 , |ṙ(t)|2 ,r(t) · r̈(t) and r(t) × ṙ(t). 1.2. INTEGRAL CALCULUS 15 1.2.7.5 Ex: Surface integrals 3 2 2 z RGiven be the vector field A = zyêx + y sin xêy + xy e êz. Calculate the integrals A · dF over the triangle (0, 0, 0) → (0, 3, 0) → (0, 0, 3) → (0, 0, 0), and over the rectangle (2, 2, 0) → (2, 4, 0) → (4, 4, 0) → (4, 2, 0) → (2, 2, 0). 1.2.7.6 Ex: Surface integrals Calculate the integral over a closed surface a. of the field A = r over the cube 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1 e b. of the field A = ρE⃗ρ over the radial surface of the cylinder 0 ≤ z ≤ 1, 0 ≤ ρ ≤ 1. 1.2.7.7 Ex: Surface integrals Calculate for the vector field A(r) = cr with c =constant the surface integral Z I = A(r) × dS F a. over the surface of a sphere (radius R, center in the origin of coordinates) b. over the surface of a cylinder (radius R, length L). 1.2.7.8 Ex: Surface integrals Prove the relationship: Z 4π 4 tij ≡ df xi xj = a δij , 3 O(a) where i, j = 1, 2, 3, x1 = x, x2 = y, x3 = z, and the integral has to be calculated on the surface of a sphere with radius a. 1.2.7.9 Ex: Volume integrals Calculate the volume integral of the function Φ = z 2 over the tetrahedron with the corners in (0, 0, 0), (1, 0, 0), (0, 1, 0), and (0, 0, 1). 1.2.7.10 Ex: Fundamental theorem for gradients What is the energy gain within the potential Φ(r) = Φ0 sinkrkr along a path keeping a constant distance from the origin. 1.2.7.11 Ex: Stokes integral theorem Calculate for following field,   yz A(r) = azx xy H using Stokes’ law the path integral A · r for an integration along a circle with radius R around the z-axis at the position z = h. 16 CHAPTER 1. FOUNDATIONS AND MATHEMATICAL FORMALISM 1.2.7.12 Ex: Stokes integral theorem H Calculate the integral C x(êx + êy ) dr, where C be the unit circle in the x-y-plane. 1.2.7.13 Ex: Gauß integral theorem Calculate the flux of the vector field A(r) = r through a sphere with radius R a. by the surface integral and b. with the help of Gauß’s theorem for the volume integral over the divergence. 1.2.7.14 Ex: Gauß integral theorem Let F be the surface of an arbitrary volume V. Determine for A(x1 , x2 , x3 ) = (ax1 , bx2 , cx3 ) the validity of the relationship, I dF · A = (a + b + c)V. F 1.2.7.15 Ex: Gauß integral theorem Be a a scalar field and B a vector field. Show, Z Z Z 3 d r B · ∇a = aB · dF − d3 r a∇ · B. V O(V ) V 1.2.7.16 Ex: Gauß integral theorem H Calculate the integral CHx(êx + êy ) · dr about a unitary circular path C along the equator and the integral F x(êx + êy ) · dF, where F be the surface of a unit sphere. 1.3 Curvilinear coordinates The most commonly used coordinate systems are Cartesian, cylindrical and spherical. Cylindrical coordinates are expressed in terms of Cartesian coordinates by,     x ρ cos ϕ y  =  ρ sin ϕ . (1.43) z z And spherical coordinates are expressed in terms of Cartesian coordinates by,     x r sin θ cos ϕ y  =  r sin θ sin ϕ . (1.44) z r cos θ 1.3. CURVILINEAR COORDINATES 17 The task now is to express the differential elements (that is, line, surface and volume elements), as well as differential operators (that is, the gradient, divergent and rotation) and the Laplacian in term of curvilinear coordinates. Let us first consider the general case. The transformation from a Cartesian coor- dinate system (x, y, z) into a general, curvilinear system (u, v, w) is given by,   x(u, v, w) r ≡ y(u, v, w). (1.45) z(u, v, w) ∂r The change du r resulting from a small variation du is then du r = ∂u du and occurs in the direction of the new unit vector êu. The unit vectors of the new system, therefore, can be written as, ∂r ∂r ∂r êu = U (u, v, w) , êv = V (u, v, w) , êw = W (u, v, w) , (1.46) ∂u ∂v ∂w where −1 −1 −1 ∂r ∂r ∂r U= , V = , W =. (1.47) ∂u ∂v ∂w In the Excs. 1.3.8.1 we study transformations into cylindrical and spherical coordi- nates. 1.3.1 Differential elements in curvilinear coordinates In the following we restrict ourselves to orthogonal coordinates, where the unit vectors are perpendicular. In this case, the total differential dr has the form, ∂r ∂r ∂r du dv dw dr = du + dv + dw = êu + êv + êw , (1.48) ∂u ∂v ∂w U V W and has the length, 2 2 2 du dv dw |dr|2 = + +. (1.49) U V W The volume element is, dτ = dsu dsv dsw. (1.50) 1.3.2 Gradient in curvilinear coordinates We can now express the gradient of a scalar field Φ in orthogonal curvilinear coordi- nates, grad Φ = ∇Φ = fu êu + fv êv + fw êw , (1.51) where the fi are functions which have yet to be determined. To this end we compare the coefficients of the expressions, ∂Φ ∂Φ ∂Φ dΦ = du + dv + dw , (1.52) ∂u ∂v ∂w 18 CHAPTER 1. FOUNDATIONS AND MATHEMATICAL FORMALISM and, inserting (1.48) and (1.51), fu fv fw dΦ = dr · ∇Φ = du + dv + dw. (1.53) U V W We obtain, ∂Φ ∂Φ ∂Φ ∇Φ = U êu + V êv + W êw. (1.54) ∂u ∂v ∂w 1.3.3 Divergence in curvilinear coordinates Now we will show how to express the divergence of a vector field A in orthogonal curvilinear coordinates, div A = ∇ · A. (1.55) The derivation is a bit complicated. We begin by expressing the unit vectors êu , êv , and êw by the gradients ∇u, ∇v, and ∇w, using the expression for the gradient (1.54), ∇u = Uêu , ∇v = V êv , ∇w = W êw. (1.56) We now express each unit vector as the vector product of two of these gradients, ∇u × ∇v = U V êu × êv = U V êw (1.57) ∇v × ∇w = V W êv × êw = V W êu ∇w × ∇u = W U êw × êu = W Uêv. After that we write A = au êu + av êv + aw êw and start considering the first term of the divergence: a u ∇ · (au êu ) = ∇ · ∇v × ∇w (1.58) VW au au = (∇v × ∇w) · ∇ + ∇ · (∇v × ∇w) using ∇ · (αA) = A · (∇α) + α(∇ · A) VW VW ∂ au ∂ av ∂ aw = V W êu · êu U + êv V + êw W ∂u V W ∂v V W ∂w V W au + [∇w · (∇ × ∇v) − ∇v · (∇ × ∇w)] using ∇ · (A × B) = B · (∇ × A) − A · (∇ × B) VW ∂ au = UV W using ∇ × (∇α) = 0. ∂u V W Similarly we can show, ∂ av ∂ aw ∇ · (av êv ) = U V W and ∇ · (aw êw ) = U V W. (1.59) ∂v U W ∂w U V With this we finally get, ∂ au ∂ av ∂ aw ∇ · A = UV W + +. (1.60) ∂u V W ∂v U W ∂w U V 1.3. CURVILINEAR COORDINATES 19 1.3.4 Rotation in curvilinear coordinates Now we will show how to express the rotation of a vector field A in orthogonal curvilinear coordinates, rot A = ∇ × A. (1.61) We write again, A = au êu + av êv + aw êw and start considering the first term of the rotation: a u ∇ × (au êu ) = ∇ × ∇u using (1.56) (1.62) a U u au = ∇ × ∇u + (∇ × ∇u) using ∇ × (αA) = (∇α) × A + α(∇ × A) Ua U u =U ∇ × êu and ∇ × (∇α) = 0 using (1.56) U ∂ au ∂ au ∂ au = U êu U + êv V + êw W × êu ∂u U ∂v U ∂w U ∂ au ∂ au = U êv W − êw V ∂w U ∂v U 1 ∂ au 1 ∂ au = U V W êv − êw. V ∂w U W ∂v U Similarly we can show, 1 ∂ av 1 ∂ av ∇ × (av êv ) = U V W êw − êu (1.63) W ∂u V U ∂w V 1 ∂ aw 1 ∂ aw ∇ × (aw êw ) = U V W êu − êv. U ∂v W V ∂u W With this we finally obtain, ∂ aw ∂ av ∂ au ∂ aw ∇ × A = êu V W − + êv U W − (1.64) ∂v W ∂w V ∂w U ∂u w ∂ av ∂ au + êw U V − , ∂u V ∂v U or, written as a determinant,  êu êv êw  U V W ∇ × A = UV W det  ∂u ∂ ∂ ∂v ∂  ∂w. (1.65) au av aw U V W 1.3.5 Cylindrical coordinates Let us now identify the general coordinates u, v, and w with the cylindrical coordinates ρ, θ, and ϕ defined in Eq. (1.43). In Exc. 1.3.8.2 we calculate for the line element, dr = dρêρ + ρdϕeϕ + dzêz , (1.66) the distance element, |dr|2 = (dρ)2 + (ρdϕ)2 + (dz)2 , (1.67) 20 CHAPTER 1. FOUNDATIONS AND MATHEMATICAL FORMALISM the surface element given by z = z(ρ, ϕ), ∂z 1 ∂z ds = − êρ − êϕ + êz ρdρdϕ , (1.68) ∂ρ ρ ∂ϕ and the volume element, dτ = rdzϕdr. (1.69) In the Excs. 1.3.8.3 to 1.3.8.5 we calculate, respectively, the gradient, ∂Φ 1 ∂Φ ∂Φ ∇Φ = êρ + êϕ + êz , (1.70) ∂ρ ρ ∂ϕ ∂z the divergence, 1 ∂ 1 ∂ ∂ ∇·A= [ρ aρ ] + [aϕ ] + [az ] , (1.71) ρ ∂ρ ρ ∂ϕ ∂z the rotation, 1 ∂az ∂aϕ ∂aρ ∂az 1 ∂ ∂aρ ∇ × A = êρ −ρ + êϕ − + êz (ρaϕ ) − (1.72) ρ ∂ϕ ∂z ∂z ∂ρ ρ ∂ρ ∂ϕ and the Laplace operator, 1 ∂ ∂Φ 1 ∂2Φ ∂2Φ ∆Φ ≡ ∇ · (∇Φ) = ρ + +. (1.73) ρ ∂ρ ∂ρ ρ2 ∂ϕ2 ∂z 2 in cylindrical coordinates. Figure 1.6: Illustration of Cartesian, polar, cylindrical and spherical coordinates. 1.3.6 Spherical coordinates Let us now identify the general coordinates u, v, and w with the spherical coordinates r, θ, and ϕ defined in Eq. (1.44). In Exc. 1.3.8.2 we calculate for the line element, dr = drêr + rdθeθ + r sin θdϕêϕ , (1.74) 1.3. CURVILINEAR COORDINATES 21 the distance element, |dr|2 = (dr)2 + (rdθ)2 + (r sin θdϕ)2 , (1.75) the surface element given by r = r(θ, ϕ), 1 ∂r 1 1 ∂r ds = êr − êθ − êϕ r2 sin θdθdϕ , (1.76) r ∂θ r sin θ ∂ϕ and the volume element, dτ = dsu dsv dsw = r2 sin θdθdϕdr. (1.77) In the Excs. 1.3.8.3 to 1.3.8.5 we calculate, respectively, the gradient, ∂Φ 1 ∂Φ 1 ∂Φ ∇Φ = êr + êθ + êϕ , (1.78) ∂r r ∂θ r sin ϕ ∂ϕ the divergence, 1 ∂ 2 1 ∂ 1 ∂ ∇·A= [r ar ] + [sin θ aθ ] + [aϕ ] , (1.79) r2 ∂r r sin θ ∂θ r sin θ ∂ϕ the rotation, 1 ∂ ∂ 1 ∂ ∂ ∇ × A = êr (sin θaϕ ) − (aθ ) + êθ (ar ) − sin θ (raϕ ) r sin θ ∂θ ∂ϕ r sin θ ∂ϕ ∂r 1 ∂ ∂ + êϕ (raθ ) − (ar ) (1.80) r ∂r ∂θ and the Laplace operator, 1 ∂ ∂Φ 1 ∂ ∂Φ 1 ∂2Φ ∆Φ ≡ ∇ · (∇Φ) = r2 + sin θ + 2 ∂ϕ2 , (1.81) r2 ∂r ∂r 2 r sin θ ∂θ ∂θ r2 sin θ in spherical coordinates. We note that the radial part of the Laplace operator can also be written, 1 ∂2 (rΦ). (1.82) r ∂r2 1.3.7 Differential operators for tensor fields Until now, we restricted to scalar and vector fields, that is spatially dependent physical quantities such as the temperature or the magnetic field distribution in a room. Some quantities, however, need to be given as matrices or even higher-dimensional objects, for example, gravity gradients or the susceptibility of a crystal. These objects are called tensors. With the definition, êk ⊗ êl = êk ê⊺l (1.83) we can express a scalar, vector, and second-order tensor as, Φ , A = Ak êk , G = Gkl êk ⊗ êl. (1.84) 22 CHAPTER 1. FOUNDATIONS AND MATHEMATICAL FORMALISM For instance, in Cartesian coordinates a second-order tensor reads, ⊺ ⊺ ⊺ ⊺ Gxx Gxy G = Gxx êx êx + Gxy êx êy + Gyx êy êx + Gyy êy êy =. (1.85) Gxy Gyy If a tensor T = Φ(r), A(r), G(r) varies in space, we can apply differential operators to it. The gradient ∇T (r) of a tensor field in the direction of an arbitrary constant vector x is defined as, d x · ∇T = lim T (r + αx). (1.86) α→0 dα The gradient of a tensor field of order n is a tensor field of order n + 1. In Cartesian coordinates, x = xêx + yêy + zêz , ∂ ∂x Φ ∇Φ = êk Φ ≡ Φ,k êk = (1.87) ∂xk ∂y Φ ∂ ∂x Ax ∂y Ax ∇A = êk ⊗ Al êl ≡ A,kl êk ⊗ êl = ∂xk ∂x Ay ∂y Ay ∂ ∂x Gxx Gxy ∇G = êk ⊗ Glm êl ⊗ êm ≡ G,klm êk ⊗ êl ⊗ êm =. ∂xk ∂y Gxy Gyy The divergence of a tensor field T (r) is defined using the recursive relation, (∇ · A) · x = Tr (∇A) (1.88) ⊺ (∇ · G) · x = ∇ · (x · G ). In Cartesian coordinates the divergence is, ∂ ∂Ak ∇ · A = êk · Al êl = ≡ Ak,k (1.89) ∂xk ∂xk ∂ ∂Glk ∇ · G = êk · Glm êl ⊗ êm = ⊗ êl ≡ Glk,k êl. ∂xk ∂xk The curl of an order n > 1 tensor field T (r) is also defined using the recursive relation, (∇ × A) · x = ∇ · (A × c) (1.90) (∇ × G) · x = ∇ × (x · G ⊺ ) , where c is an arbitrary constant vector. In Cartesian coordinates the divergence the rotation is, ∇ × A = ϵklm êk ∂l Am (1.91) ∇ × G = ϵklm êm ⊗ ên ∂k Gnl. 1.3.8 Exercises 1.3.8.1 Ex: Spherical and cylindrical coordinates a. Express the cylindrical coordinates ρ, φ, z in terms of the Cartesian ones x, y, z. b. Express the spherical coordinates r, θ, φ in terms of the Cartesian ones x, y, z. 1.3. CURVILINEAR COORDINATES 23 1.3.8.2 Ex: Differential elements in curvilinear coordinates We have seen in class that the transformation from a Cartesian coordinate sys- tem (x, y, z) to another curvilinear and orthogonal system (u, v, w) is given by r ≡ (x(u, v, w), y(u, v, w), z(u, v, w)). Now consider the spherical polar coordinates r ≡ (x(r, θ, ϕ), y(r, θ, ϕ), z(r, θ, ϕ) defined in class. a. Calculate the functions Ur , Vθ , Wϕ defined by, −1 −1 −1 ∂r ∂r ∂r Ur = , Vθ = , Wϕ =. ∂r ∂θ ∂ϕ b. Determine the Cartesian coordinates of the new unit vectors êr , êθ , êϕ , draw the position of these vectors at a point r0 , and check the orthogonality of the unit vectors. Express the Cartesian unit vectors by the spherical ones. c. Determine the total differential dr, the line element (ds)2 = |dr|2 , and the volume element dτ = dsu dsv dsw in terms of the new coordinates. d. Repeat steps (a)-(c) for planar polar coordinates. 1.3.8.3 Ex: Spherical and cylindrical coordinates Calculate ∇Φ, ∇ · A, ∇ × A and ∆Φ = ∇ · (∇Φ) a. in spherical coordinates (r, θ, ϕ). b. in cylindrical coordinates (ρ, ϕ, z). 1.3.8.4 Ex: Divergence in curvilinear coordinates  2  x y Calculate the divergence of the force field F(r) =  2yz  (a) in Cartesian and (b) x+z cylindrical coordinates and compare the results. 1.3.8.5 Ex: Differential operators in curvilinear coordinates In Cartesian coordinates the differential line element has the form dr = êx dx+êy dy + êz dz and in arbitrary orthogonal coordinates dr = ê1 h1 dq1 + ê2 h2 dq2 + ê3 h3 dq3 −1 ∂r ∂r ∂r with êi = ∂qi · ∂qi and hi = ∂qi. For spherical coordinates (r, ϕ, θ) we find hr = 1, hϕ = r sin θ, hθ = r; for cylindrical coordinates (ρ, ϕ, z) we find hρ = 1, hϕ = ρ, hz = 1. a. The gradient has the general form, 1 ∂ ∇i Φ(r) = Φ(r). hi ∂qi Determine the gradient in spherical and cylindrical coordinates. b. The divergence of a vector field A has the general form, 1 ∂ ∂ ∂ ∇ · A(r) = (A1 h2 h3 ) + (A2 h1 h3 ) + (A3 h1 h2 ). h1 h2 h3 ∂q1 ∂q2 ∂q3 24 CHAPTER 1. FOUNDATIONS AND MATHEMATICAL FORMALISM Determine ∇ · A in spherical and cylindrical coordinates. c. Use the results of (a) and (b) to determine the Laplace operator ∆=∇·∇ in spherical coordinates. 1.3.8.6 Ex: Acceleration in spherical coordinates In spherical coordinates the velocity vector has the following form, dr v= = ṙêr + rθ̇êθ + ϕ̇r sin θêϕ. dt Calculate the acceleration vector in spherical coordinates. Respect the fact that the basis vectors must also be derived by time. 1.3.8.7 Ex: Volume element in curvilinear coordinates a. Calculate the surface of a rectangle with width a and height b in Cartesian coordi- nates. b. Calculate the surface of a disk of radius R in polar coordinates. c. Calculate the volume of a cuboid with dimensions a, b, c in Cartesian coordinates. d. Calculate the volume of a cylinder with the radius R and height H in cylindrical coordinates. e. Calculate the volume a sphere with the radius R in spherical coordinates. Figure 1.7: Curvilinear coordinates. 1.3.8.8 Ex: Spherical volume The volume of a body is given by the following formula: Z V = 1 dV. V 1.3. CURVILINEAR COORDINATES 25 a. Calculate the volume of a 3D-sphere in spherical coordinates. By the Gauß integral law we can establish a relationship between the volume of the sphere and its surface. (Help: For which vector field A holds: ∇ · A = 1?) b. Now calculate the volume of the sphere in this sense. You may assume that the surface of the sphere is known. c. Similarly to the above formula, derive a general relationship between the volume of an n-dimensional hypersphere and its (n − 1)-dimensional hypersurface. Help: Gauß’s law holds in arbitrary dimensions with the n-dimensional operator nabla- operator defined by, ∇ = (∂/∂x1 ,... , ∂/∂xn ). 1.3.8.9 Ex: Spherical volume 2 2 2 The density distribution of a gas be given by n(r) = C 2 − xr2 − yr2 − zr2. Determine the 0 0 0 constant C in such R a way3 that the density n(r) is normalized to the number of atoms in the gas, i.e., V n(r)d r = N , where V is the volume within which the density is positive, n(r) ≥ 0. 1.3.8.10 Ex: Spherical and cylindrical volume a. Integrate a circular surface with radius R in Cartesian coordinates and then in polar coordinates. 2 2 b. The density distribution of a trapped atomic gas is described by n(r) = n0 e−r /r̄ , where n0 = 1013 cm-1 is the maximum density and r̄ = 100 µm R a measure for the ex- tent of the distribution. Calculate the number of atoms N = R3 n(r)d3 r, integrating over Cartesian coordinates and then over polar coordinates. c. Calculate the density of a homogeneous cylinder of mass 10 kg and length 20 cm by integrating over its volume.. d. The densityndistribution of a trapped o atomic gas is described by 2 2 n(ρ, z) = max 0, n0 · 1 − ρρ2 − zz2 , where n0 = 1013 cm-3 is the maximum den- m m sity and zm = 2ρm = 100 µm a measure for the extent of the distribution. Calculate the number of atoms by integrating over cylindrical coordinates. 1.3.8.11 Ex: Cylindrical volume Consider a material (gas or liquid) whose mass density ρ(r) depends on the z- coordinate as follows: ρ(r) = ρ0 (1 − αz). This material is filled into a cylinder (radius R and height c) until the total mass in the cylinder is M. The cylinder stands in a circular area above the xy in z = 0. a. Calculate the density parameter ρ0. b. Calculate vector of center of mass rs of the material in the cylinder. c. Now fill the same material inside a sphere of radius R instead of the cylinder. What are the results in this case if α = 0.1 /m, R = 1 m, and M = 10 kg. d. A cake of mass M , height h and radius R be cut into fourth equal pieces. Calculate the center of mass of a piece. Calculate the center of mass of the rest of the cake when a piece is taken. 26 CHAPTER 1. FOUNDATIONS AND MATHEMATICAL FORMALISM 1.3.8.12 Ex: Vector potential in curvilinear coordinates Be given a constant field B oriented in z-direction. What is the vector potential A in (a) spherical coordinates, (b) cylindrical coordinates, and (c) Cartesian coordinates? For case (c) also consider the gauge transformation A′ = A + ∇λ with λ = ±Bxy/2. 1.3.8.13 Ex: Gauß’ theorem in curvilinear coordinates a. Check Gauß’ theorem for function A = r2 êr using the volume of a sphere of radius R. b. Do the same for the function B = r−2 êr and discuss the result. 1.3.8.14 Ex: Gauß’ theorem in curvilinear coordinates Calculate the divergence of the function, A = rêr cos θ + rêθ sin θ + rêφ sin θ cos φ. Verify Gauß’s theorem for this function using the volume of an inverted semisphere of radius R lying in the x-y-plane. 1.3.8.15 Ex: Gauß’ theorem in curvilinear coordinates Calculate the gradient and Laplacian of the function T = r(cos θ + sin θ cos φ). Check the Laplacian by converting T into Cartesian coordinates. Verify Gauß’s theorem using the path l1 (t) = 2êx cos πt + 2êy sin πt for t ∈ [0, 0.5] followed by l2 (t) = 2êy sin πt − 2êz cos πt para t ∈ [0.5, 1]. 1.3.8.16 Ex: Gauß’ theorem in curvilinear coordinates a. Find the divergence of the function, A = ρêρ (2 + sin2 φ) + ρêφ sin φ cos φ + 3zêz. b. Verify Gauß’ theorem for this function using a quadrant of cylinder with radius R = 2 and height h = 5. c. Find the rotation of A. 1.4 Differential geometry in curved space In previous sections we mainly concentrated on orthogonal coordinate systems, such as Cartesian, cylindrical, or spherical. In cases one has to use non-orthogonal systems the formalism needs to be generalized. For this purpose it is necessary to introduce some new concepts and notations. Pn Repeat expressions in index formalism Einstein’s sum rule, an an = a an , ds2 = gmn dxm dxn. (1.92) 1.4. DIFFERENTIAL GEOMETRY IN CURVED SPACE 27 1.4.1 Co- and contravariant tensors A contravariant vector or tangent vector (often abbreviated simply as vector, such as a direction vector or velocity vector) has components that contra-vary with a change of basis to compensate. That is, the matrix that transforms the vector components must be the inverse of the matrix that transforms the basis vectors. Examples of contravariant vectors include the position of an object relative to an observer, or any derivative of position with respect to time, including velocity and acceleration. Contravariant components are denoted with upper indices as in, v = v i ei. (1.93) A covariant vector or cotangent vector has components that co-vary with a change of basis. That is, the components must be transformed by the same matrix as the change of basis matrix. Examples of covariant vectors generally appear when taking a gradient of a function. Covariant components are denoted with lower indices as in, w = wi ei. (1.94) Figure 1.8: (a) Covariant and (b) contravariant basis. 1.4.2 Jacobian for coordinate transformations The Jacobian of a vector field F is a matrix defined by, ∂(F1 ,.., Fm ) ∂Fm J≡ ≡ ≡ (J mn ). (1.95) ∂(x1 ,.., xn ) ∂xn We may understand a curvilinear coordinate system {xµ } as a vector field in Cartesian space {x′ν }, xµ = xµ (x′1 ,.., x′ν ,.., x′m ) (1.96) for µ = 1,.., m. The Jacobi

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