Rope around the earth.
Gamma function, Beta function.
Approximation of integrals, Stirling's approximation for n!
||Begin Chapter 4. Sequences and series.
Tests of convergence. Preliminary test. Comparison test.
||Sums of series. Integral test. Ratio test.
Geometric series. Logarithmic series. Power law power series.
Expansions and limits in physics: Blackbody radiation.
||Labor Day (no class)
Chapter 5, partial derivatives.
Changing variables, expanding universe.
Thermodynamics and Maxwell relations.
||Hurricane Irma! (no class)
||Hurricane Irma!!! (no class)
||Hurricane Irma!! (no class)
Max/min with constraints, Lagrange multipliers.
||Begin Chapter 6. Multiple integrals.
Change of variables, Jacobian, area in hyperbolic coordinates.
Integrals along curves.
||Lengths of curves and areas of surfaces.
||Exam 1 (in class)
||Begin Chapter 7, Vectors.
Dot product, δij.
cross product, εijk.
Vector derivatives, product rules.
Scalar, vector fields.
Vector derivatives, gradient, divergence, curl.
Second derivatives, Laplacian.
||Curl of a cross product.
Gradient, divergence, Laplacian in polar (cylindrical) coordinates.
||Chapter 11, integrals.
Divergence Theorem, Stokes's Theorem.
Equation of continuity.
||Flux, divergence for point charge:
Properties of δ-function.
||Homecoming (no class)
Regularization of point charge.
Electromagnetism, charge conservation, electromagnetic waves.
||Chapter 3, Complex numbers. Euler's formula.
||Damped harmonic oscillator,
Chapter 24, Complex functions.
||Cauchy's theorem. Cauchy's formula.
||Applying Cauchy's formula:
ζ(2) as a contour integral.
||Complex contour integrals with branch cuts
||Matrix algebra, multiplication, determinant, inverse, transpose, trace.
||Exam 2 (in class)
||Eigenvalues: vibrational modes of a molecule.
Determinant, Trace invariants.
||Schwarz inequality, uncertainty principle.
||Eigenvalues and eigenvectors of self-adjoing linear differential operator,
L = -i d/dθ.
||Fourier series for square wave; convergence, overshoot.
||Veteran's Day (no class)
||Integral of square wave = triangle wave.
Derivative of square wave =
δ-function, partial sum.
||L → ∞,
Lorentzian, Gaussian examples.
||Fourier transform of "top hat".
Convolution. Parseval's theorem.
Aside on Laplace transform, Mellin transform.
||Fourier transform solution of damped harmonic oscillator.
Fourier transform solution of Poisson's equation.
||Thanksgiving (no class)
Bessel equation. Orthogonal polynomials, Hermite polynomials.
Hermite series for sin(x),
Taylor series for sin(x)
||Partial differential equations,
Laplace/Poisson equation, wave equation, Schrö'dinger equation,
Separation of variables in Cartesian, spherical coordinates.
||Helmholtz equation, spherical waves, scattering,
||Exam 3 (in class)