## Sunday, January 29, 2012

### Ben's Physics Problem Archive

I've imported my homework assignments from graduate school into Google Docs, and they are now available in the link to the right.  Their contents are catalogged are in the readme file at the top of the list for easy lookup.

These digitized notes have proven incredibly useful to me, and I provide them here so that others might benefit for them as well.  Please do not copy from them for your homework assignments or exams, as that will only harm you in the long term.

Links to PDFs and my original descriptions of the assignments can be found below.

## Supplemental Problem Archive

 Link Source Description Special Topics in Statistical Physics Exact solution to the Ising model in 1 dimension Special Topics in Statistical Physics Properties of Van Der Waals fluids near critical temperature Biophysics 33-767 Pulling forces on stretched or nearly-stretched polymers and pulling forces on looped polymers. Biophysics 33-767 Viral Budding using the shape equation on a membrane and a spherical colloid particle. Biophysics 33-767 Toroidal vesicles, the Helfrich Hamiltonian, and the shape equation as the linearized term of the Helfrich Hamiltonian. Biophysics 33-767 Calculation of the time autocorrelation function in Fluorescence Correlation Spectroscopy, and the scaled volume fraction of N-aggregates in cylindrical micelles. Biophysics 33-767 The effects of dimensionality of space on a particle's locality under diffusion, a derivation of the Stokes-Einstein relation, and the solubility of oil in water based on the length of the aliphatic chain of the hydrocarbon, in the ideal-gas approximation. Biophysics 33-767 A proof that entropy increases under the diffusion equation in arbitrary dimensions, the time-evolution of a Gaussian distribution under diffusion, and the time-evolution of a square wave under diffusion. Jackson, Ficenec and Teplitz A research outline for Magnetic Monopoles that constituted my notes for the oral qualifying exam. Includes: The Dirac Quantization Condition, Duality Transformations, Dirac Strings, Coulomb's Law for Monopoles, Gauge Transformations, Anomalous Hyperfine Splitting in the Hydrogen Atom, Ionization losses in matter, Nuclear, Atomic and Ferromagnetic binding in matter, and Monopoles in the early universe, the Aharanov-Bohm effect, and quantum mechanical properties of monopoles. Problems: Jackson 6.16, 6.17, 6.18, 6.19 Ficenec and Teplitz Chapter 8 Problems 1-12, 15-22 Original A simple example and intuitive explanation as to why charge on a conductor in equilibrium in free space tends to accumulate on areas with a smaller radius of curvature. Griffiths Quantum Mechanics Problem 4.44 Griffiths 4.44 The expectation value of an unusual operator acting on the singlet state (composed of two spin-half particles). Here you see a brute-force evaluation of the operator that underlies this problem. The procedure at this level is surprisingly simple, but the definitions that underlie it might be intimidatingly complex. In fact, simple substitution of the definitions of the underlying the state and operators ultimately leads to an ugly but straightforward evaluation.

## Semester 4 Problem Archive

 Link Source Description Numerical Analysis Homework 6 Stability Analysis, the Crank-Nicholson Method, Forwards Euler, Central Euler, and Backwards Euler, plus a Mathematica implementation each for the heat equation, the wave equation, and a velocity equation. Kincaid 9.2.2, 9.7.4, 9.7.5 Continuum Mechanics Homework 5 Constitutive classes, ideal fluids, Newtonian fluids, Reiner-Rivlin fluids, independence of observer, the First Representation Theorem for Isotropic Tensor Functions, flows of Reiner-Rivlin fluids, Response functions, and the Piola-Kirchoff stress tensor. Gurtin 16.1, 19.2, 21.1, 22.1, 24.1, 25.2, 27.2, 27.5 Numerical Analysis Computer Project 2 A Mathematica implementation of the Richardson, Jacobi, Gauss-Seidl and the Kaczmarz algorithms, used to study an analog of Simpson's Rule. Continuum Mechanics Homework 4 Angular momentum, rigid motions, inertia tensors, Euler's Equations, Normal stresses, Signorini's Theorem. Gurtin 13.2, 13.3, 14.1, 14.4, 14.5, 14.9, 15.1, 15.3 Continuum Mechanics Homework 3 Identities on simple shears, velocity fields, streamlines, motions under a shift in reference time, Rivlin-Erickson tensors, and potentials. Gurtin 8.1, 8.3, 8.4, 8.5, 9.1, 9.4, 11.2, 11.5 The Rayleigh Instability A simple derivation of the Rayleigh Instability for a fluid cylinder. Numerical Analysis Homework 5 Runge-Kutta formulas, the modified Euler's method, and autonomous systems of first-order equations. Kincaid 8.3.3, 8.3.4, 8.3.5, 8.3.6, 8.6.6 Numerical Analysis Homework 4 Richardson Extrapolation, approximating derivatives, numerical integration rules, Gaussian quadrature, the method of undetermined coefficients, the Newton-Cotes formula, and Euler-Maclaurin formulas. Kincaid 7.1.3, 7.1.6, 7.1.12, 7.1.14, 7.1.15, 7.2.4, 7.2.5, 7.2.8, 7.2.12, 7.2.13, 7.2.20, 7.2.23, 7.3.11, 7.3.15, 7.3.17, 7.3.21, 7.3.25, 7.3.31, 7.4.1, 7.4.2 Continuum Mechanics Homework 2 Component representations, identities on curl and divergence, the divergence theorem, plane strains, isochoric flows, and Korn's inequality. Gurtin 4.3, 4.10, 5.1, 5.2, 6.8, 6.10, 7.1, 7.3, 7.5 Continuum Mechanics Homework 1 Identities on general vector spaces, skew and symmetric tensors, spectra, characteristic spaces, polar decompositions, similarity transforms, differentiation of tensors, differentiation by Jacobians, and orthogonal tensors. Gurtin 1.6, 1.14, 1.15, 2.1, 2.3, 2.6, 3.1, 3.2, 3.6 Numerical Analysis Homework 3 Matrix norms and iterative methods for solving many-variable equations. Kincaid 4.4.1, 4.4.2, 4.4.3, 4.4.8, 4.4.11, 4.4.17, 4.4.18, 4.4.33, 4.4.49, 4.5.3, 4.5.19, 4.6.2, 4.6.12, 4.6.33, 4.6.39 Numerical Analysis Homework 2b The Newton algorithm in complex arithmetic (program, practical example), and the Secant method (program, practical example). Kincaid Computer Projects 3.3.8, 3.3.6 Numerical Analysis Homework 2a Fixed points, Continued Fractions. Kincaid 3.4.3, 3.4.6, 3.4.9, 3.4.12, 3.4.13, 3.4.40, Numerical Analysis Homework 1 Taylor Series, rates of convergence, Newtonian Iteration, Halley's Method. Kincaid 1.1.29, 1.1.32, 1.2.2, 3.1.14, 3.1.16, 3.2.14, 3.2.15, 3.2.16, 3.2.19, 3.2.21,