Teaching

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Mathematical Methods of Physics

These lecture notes for the courses PHY510/610 are a work in progress at this time. Amendments, improvements, and corrections wil be made continually. The topics listed are interrelated in intricate ways. They do not reflect a sequence of course materials. The lectures visit each topic multiple times.

Complex Analysis 

• Complex numbers [gmd7-A]
• Stereographic projection
• Complex functions
• Complex derivatives
• Differential operators
• Orthogonal families of curves

• Line integrals [gmd7-B]
• Cauchy’s theorem
• Cauchy integrals
• Taylor series
• Laurent series
• Residues
• …

Exercises:

• Trigonometric relations made transparent by complex variables [gex16]
• Search for zeros of functions in the complex plane I [gex18]
• Constructing inverse trigonometric functions I [gex19]
• Constructing analytic function from harmonic function [gex21]
• Derivatives of inverse trigonometric functions [gex22]
• Harmonic and analytic functions I [gex54]
• Applications of L’Hospital’s rule [gex55]
• Orthogonal family of functions I [gex56]
• Conducting plates intersecting at right angle [gex57]
• Electric potential and field at edge of large conducting plate [gex58]
• Fringe electric potential and fringe field of parallel plates [gex59]
• Differential operators acting on complex functions I [gex60]
• Line integral in complex plane I [gex77]
• Green’s theorem adapted to complex functions [gex78]
• Application of Cauchy’s theorem I [gex79]
• Argument theorem in complex analysis [gex80]
• Poisson integrals for circle in complex plane [gex81]
• Application of Cauchy’s theorem II [gex82]
• Poisson integrals for half plane in complex analysis [gex83]
• Applications of Poisson integrals in complex analysis I [gex84]
• Laurent series of analytic functions I [gex85]
• Laurent series of analytic functions II [gex86]
• Laurent series of analytic functions III [gex87]
• Residues of isolated singularities of complex functions [gex88]
• Definite integral via residue theorem I [gex89]
• Definite integral via residue theorem II [gex90]
• Definite integral via residue theorem III [gex91]
• …

Additional Materials:

• Complex functions in electroststics [gam1]
• …

Special functions 

• Elementary functions [gmd4]
• Special functions

• Gamma function [gmd4A]
• Beta function
• Incomplete Gamma function
• Binomial series

• Error function [gmd4B]
• Fresnel integrals
• Elliptic integrals and elliptic functions [gmd4C]
• Legendre polynomials and functions [gmd4D]
• Spherical harmonics [gmd4E]

• Bessel functions [gmd4F]
• Modified Bessel functions
• …

Exercises:

• Recurrence relation for Gamma function [gex2]
• Relation between Gamma and Beta function [gex3]
• Complete elliptic integrals: series expansion [gex8]
• Area and circumference of an ellipse [gex9]
• Plane-pendulum oscillations [gex10]
• Plane-pendulum rotations [gex11]
• Electric field of charged ring and elliptic integrals [gex34]
• Magnetic field of circular current and elliptic integrals [gex35]
• Duplication relation between Gamma for Gamma function [gex39]
• Euler’s product representation of Gamma function [gex53]
• Polygamma function: series representation, recurrence relation [gex92]
• Polygamma function: integral representation, asymptotic series [gex93]
• Complementary error function: asymptotic expansion [gex94]
• Relations between error function and Fresnel integrals [gex95]
• …

Additional Materials:

• …

Matrix Operations 

• Fundamental properties, operations, and relations [gmd6-A]
• Determinants 
• Inverse matrix
• Orthogonality and unitarity
• Systems of linear equations
• Eigenvalues and eigenvectors
• Mathematica notebook [gmd6-A.nb]

Exercises:

• Matrix operations I: matrix multiplication [gex40]
• Matrix operations II: Laplace expansion of determinant [gex41]
• Matrix operations III: inverse square matrix [gex42]
• Matrix operations IV: orthogonal matrix [gex43]
• Matrix operations V: unitary matrix [gex44]
• Matrix operationss VI: determinant of matrix product [gex45]
• Matrix operations VII: system of linear equations [gex46]
• Matrix operations VIII: eigenvectors of symmetric matrix [gex47]
• Matrix operations IX: eigenvectors of Hermitian matrix [gex48]
• Matrix operations X: eigenvectors of orthogonal matrix [gex49]
• Matrix operations XI: eigenvectors of unitary matrix [gex50]
• Matrix operations XII: eigenvectors of asymmetric matrix [gex51]
• Matrix operations XIII: eigenvectors of transition matrix [gex52]
• …

Additional Materials:

• …

Vector Analysis 

• Vector addition [gmd1-A]
• Dot product of vectors
• Cross product of vectors
• Triple product of vectors
• Reciprocal vectors
• Vector functions
• Scalar and vector fields
• Differential operators
• Identities involving differential operators
• Differentials of scalars
• Differentials of vectors
• Mathematica notebook [gmd1-A.nb]

• Vector integrations [gmd1-B]
• Line integrals
• Surface integrals
• Differential operators from integrals
• Integral theorems
• Green’s identities
• Integration by parts generalized
• Helmholtz theorem
• …

Exercises:

• Electric dipole field [gex26]
• Magnetic dipole field [gex27]
• Reciprocal vectors [gex28]
• Expansion of vector in non-orthonormal basis [gex29]
• Diagonals of a parallelogram [gex30]
• Law of sines for spherical triangle [gex32]
• Common identities used in electrostatics and elsewhere [gex33]
• Vector divisions? [gex36]
• Gradients of related scalar functions [gex37]
• Work done by conservative force I [gex63]
• Work done by conservative force II [gex64]
• Electrostatic field of two point charges [gex65]
• Electrostatic field of three point charges [gex66]
• Vector functions I [gex124]
• …

Additional Materials:

• …

Coordinate Systems 

• General coordinates [gmd2-A]
• Orthogonal coordinates
• Gradient, divergence, curl, and Laplacian
• Determination of scale factors
• Cartesian coordinates
• Cylindrical coordinates
• Spherical coordinates

• Parabolic cylindrical coordinates [gmd2-B]
• Paraboloidal coordinates
• Elliptic cylindrical coordinates
• Prolate spheroidal coordinates
• Oblate spheroidal coordinates
• Ellipsoidal coordinates
• …

Exercises:

• Vector fields in curvilinear coordinates I [gex67]
• Vector fields in curvilinear coordinates II [gex68]
• Vector fields in curvilinear coordinates III [gex69]
• Laplacian operating on vector field [gex70]
• Parabolic cylindrical coordinates [gex71]
• Paraboloidal coordinates [gex72]
• Elliptic cylindrical coordinates [gex73]
• Prolate spheroidal coordinates [gex74]
• Oblate spheroidal coordinates [gex75]
• Ellipsoidal coordinates [gex76]
• …

Additional Materials:

• …

Tensor Analysis 

• Introduction [gmd5-A]
• Inertia tensor
• From matrices to tensors
• Tensors in real coordinate space
• Contravariance versus covariance
• Invariance
• Mixed variance
• Affine tensors
• Cartesian tensors

• Tensor operations [gmd5-B]
• Quotient rule
• Arrays of elements — vectors, matrices, tensors
• …

Exercises:

• Inertia tensor from momentum of rigid body [gex96]
• Inertia tensor from rotational kinetic energy of rigid body [gex97]
• Tensor summation convention I [gex98]
• Tensor summation convention II [gex99]
• Tensor summation convention III [gex100]
• From polar to rectangular coordinates and back: Jacobians [gex112]
• Uniform vector field in plane made into a tensor [gex113]
• Radial and azimuthal fields in a plane [gex117] 
• Application of quotient rule I [gex118]
• Application of quotient rule II [gex119]
• …

Additional Materials:

• Summation convention [gam2]
• …

Ordinary Differential Equations

• First-order ODEs [gmd10-A]
• Second-order ODEs
• …

Exercises:

• First-order ODE: separation of variables I [gex4]
• First-order ODE: separation of variables II [gex5]
• First-order ODE: separation of variables III [gex6]
• First-order ODE: exact differential I [gex7]
• First-order ODE: integrating factor I [gex12]
• First-order ODE: linearity [gex13]
• First-order ODE: homogeneity [gex14]
• First-order ODE: Bernoulli type [gex15]
• First-order ODE: convertibility [gex17]
• Second-order ODEs reducible to first order [gex20]
• First-order ODE: general, particular, and singular solutions [gex23]
• ODE for two-parameter family of conics [gex24]
• First-order ODE: Clairaut type [gex25]
• Second-order ODE: fixed points and isoclines I [gex101]
• Second-order ODEs: fixed points and isoclines II [gex102]
• Plane pendulum with attenuation: fixed points and phase flow [gex103] 
• Coupled first-order ODEs: fixed points and flow dynamics I [gex104]
• Coupled first-order ODEs: fixed points and flow dynamics II [gex105]
• Coupled first-order ODEs: fixed point and limit cycle [gex106]
• Coupled first-order ODEs: Rössler band strange attractor [gex107]
• Second-order ODE: reduction to first-order ODE I [gex108]
• Second-order ODE: reduction to first-order ODE II [gex109]
• Linear second-order ODE with degenerate roots [gex110]
• Linearly damped harmonic oscillator: general solution [gex111]
• Linear inhomogeneous ODE: undetermined constant parameters I [gex114]
• Linear inhomogeneous ODE: undetermined constant parameters II [gex115]
• Second-order ODE: reduction to first order III [gex116]
• …

Additional Materials:

• Linear ODEs [gam8]
• General structure of ODE and solution
• Homogeneous ODE with constant coefficients
• Linearly damped harmonic oscillator
• Particular solution of inhomogeneous ODE
• Method of undetermined constant parameters
• Method of variation of parameters

• ODEs of classical dynamical systems [gam3]
• Dynamical systems of two variables
• Isoclines
• Fixed points
• Conservative forces
• Limit cycle
• Dynamical systems of three variables
• Is classical mechanics a deterministic theory?

Integral Transforms 

• General form and types [gmd8]
• Laplace transform [gmd8A]
• Fourier transform [gmd8B]
• …

Exercises:

• Linerar ODE solved via Laplace transform [gex61]
• Laplace transform of derivatives of functions [gex62]
• …

Additional Materials:

• …

Partial Differential Equations

• Classification [gmd11-A]
• Structure og general solution according to type
• Subsidiary conditions
• …

Exercises:

• PDEs solved and solutions visualized [gex120]
• PDE solved via reduction to ODE I [gex121]
• PDE solved via reduction to ODE II [gex122]
• Laplace equation for conducting hyperbolic trough [gex123]
• …

Additional Materials:

• Laplace equation [gam4]
• Diffusion equation [gam5]
• Wave equation [gam6]
• Schrödinger equation [gam7]
• Fokker-Planck equation [gam9]
• Navier-Stokes equation [gam10]
• …

Probabilities 

• Elementary probabilities [gmd9-A]
• Elements of set theory
• Elements of probability theory
• Joint and conditional probabilities
• Statistical uncertainty and information
• …

Exercises:

• …

Additional Materials:

• …

Generalized Functions 

• Dirac delta function [gmd3-A]
• …

Exercises:

• Representations of the Dirac delta function I [gex1]
• Representation of the Dirac delta function II [gex38]
• …

Additional Materials:

• …

Functional Analysis [gmd12]

• …

Exercises:

• …

Additional Materials:

• …

Differential Geometry [gmd13]

• …

Exercises:

• …

Additional Materials:

• …

Calculus of Variation [gmd14]

• …

Exercises:

• Searching for extremum with Lagrange multiplier [gex31]
• …

Additional Materials:

• …

Group Theory and Symmetry Transformations [gmd15]

• …

Exercises:

• …

Additional Materials:

• …

Some Relevant Textbooks and Monographs:

• L. C. Andrews: Special Functions for Engineers and Applied Mathematicians. Macmillan, 1985.
• L. C. Andrews: Elementary Partial Differential Equations. Academic Press, 1986.
• L. C. Andrews and B. K. Shivamoggi: Integral Transforms for Engineers and Applied Mathematicians. Macmillan, 1988.
• G. B. Arfken and H. J. Weber: Mathematical Methods for Physicists.  Harcourt, 2001.
• R. Bronson and G. B. Costa: Differential Equations. McGraw-Hill, 2014.
• R. Courant and D. Hilbert: Methoden der Mathematischen Physik I, II. Springer, 1968.
• P. Dennery and A. Krzywicki: Mathematics for Physicists. Dover, 1995.
• G. H. Golub and C. F. van Loan: Matrix Computations. Johns Hopkins University Press, 1985.
• M. Karbach: Mathematische Methoden der Physik. De Gruyter, 2017.
• D. C. McKay: Tensor Calculus. McGraw Hill, 2011.
• D. E. Neuenschwander: Tensor Calculus for Physics. Johns Hopkins University Press, 2015
• M. R. Spiegel: Advanced Mathematics for Engineers and Scientists. McGraw-Hill, 1971.
• M. R. Spiegel, S. Lipschutz, and D. Spellman: Vector Analysis. McGraw-Hill, 2009.
• M. R. Spiegel, S. Lipschutz, J. J. Schiller,and D. Spellman: Complex Variables. McGraw-Hill, 2009.
• …

Last updated: 09/05,23