Duke Physics Course PHY231 Outline
Mathematical Methods in Electromagnetism
Numbers in parentheses are Arfken (A) and Jackson (J) sections related
to the material. See the course synopsis for
general information.
One-dimensional Green's functions
Intuitive approach.
Construction from homogeneous solutions.
Green's theorem.
Symmetry.
Eigenfunction expansion of Green's functions.
(A9.4, A16.5, J3.9)
Curvilinear Coordinates
Coordinate transformations.
Jacobian matrix.
Base vectors.
Volume element.
Metric tensor.
Orthogonal coordinates.
Scale factors.
Coordinate curves and surfaces.
Vector components.
Vector differential operators.
(A2.1-2.5)
Non-orthogonal Coordinates
Contravariant and covariant representations.
Reciprocal vectors.
Raising and lowering of indices.
Vector differential operators.
Covariant derivatives.
(A4.4)
Tensors, Vector Calculus
General tensors.
Transformation laws.
Cartesian tensors.
Tensor operations.
Isotropic tensors.
Pseudotensors.
Symmetry.
Vector calculus identities and theorems.
(A1.6-1.12, A3.1-3.4, A3.8, J-inside cover)
Fundamental Laws of Electromagnetism
Maxwell's equations.
Units and dimensions.
Laws of Coulomb, Gauss, Biot & Savart, Ampère, and Faraday.
Scalar and vector potentials.
Laplace and Poisson equations.
Covariant formulation.
Energy and power.
Poynting's theorem.
Macroscopic media.
Boundary conditions.
Electrostatic energy in media.
(JI.4-I.5, J1.1-1.7, J4.3, J4.7,
J5.1-5.4, J5.8, J6.1-6.5, J6.7-6.8, J11.9, J-appendix)
Electrostatics and Potential Theory
Survey.
Method of images.
Coulomb potentials.
Uniqueness.
Helmholtz' theorem.
3d Green's functions.
All-space and spherical Green's functions.
Eigenfunction expansion for rectangular Green's function.
Reduction to 1d Green's function.
Expansions for rectangular Laplace problems.
(A1.13-1.15, A8.7, A16.6, J1.8-1.11, J2.1-2.7, J2.9-2.11, J3.12, J5.5)
Partial Differential Equations
PDE's of physics.
Classification and boundary conditions.
Separation of variables in rectangular, spherical, and cylindrical systems.
Standard solutions of Helmholtz's equation.
(A2.6, A8.1, A8.3, A9.1, J3.1, J3.7)
Legendre Functions
Legendre polynomials.
Rodrigues' formula.
Generating function.
Recurrence relations.
Orthogonality and Legendre Series.
Expansions for axisymmetric spherical Laplace problems.
Electro/magneto-static problems.
Legendre functions of order nu.
Legendre functions of the 2nd kind.
Other orthogonal polynomials.
(A12.1-12.4, A12.10, A13.1-13.4, J3.1-3.4, J4.4, J5.9-5.12)
Spherical Harmonics
Associated Legendre functions.
Spherical harmonics.
Laplace series.
Expansions for axisymmetric spherical Laplace problems.
Completeness and 3d delta functions.
Angular momentum.
Spherical Green's functions.
Addition theorem.
2-center integrals.
Multipole expansion.
(A12.5-12.8, A16.6, J3.5-3.6, J3.9-3.10, J4.1-4.2, J5.5)
Cylindrical Bessel Functions
Overview of all Bessel functions.
Large and small x behavior.
Series solutions.
Second solutions.
Generating functions.
Recurrence relations.
Integral representations.
Hankel functions.
Integrals and diffraction.
Asymptotic forms.
Orthogonality and Bessel series.
Drumhead.
Expansions for cylindrical Laplace problems.
Cylindrical Green's functions.
(A11.1-11.6, A16.6, J3.7-3.8)
Operators with Continuous Spectra
Sturm-Liouville problems.
Hankel transforms.
Expansions for parallel-plane Laplace problems.
Parallel-plane and all-space Green's functions.
(A16.6, J2.8, J3.8, J3.11-3.12)
Spherical Bessel Functions
Raleigh formula. Orthogonality. Helmholtz resonator.
Scattering theory.
(A11.7, J3.13, J5.13, J16.1)
Last updated: 18-Sep-95