The Grammar of Nature

L04Eigenvectors and Normal ModesNext

CoversEigenvectors · the characteristic equation · diagonalization · the spectral theorem · normal modes · beats

L05Partial Derivatives and the Gradient

CoversSurfaces and slices · the total differential · the chain rule in several variables · the gradient as steepest ascent · level sets · Lagrange multipliers

L06Multiple Integrals and the Jacobian

CoversIterated integrals · change of variables as area distortion · the Jacobian · polar · cylindrical · spherical coordinates

L07Div, Grad, Curl

CoversVector fields as flows · divergence as source density · curl as local rotation · potentials and when they exist · the Laplacian

L08The Integral Theorems

CoversGauss · Stokes · Green · the fundamental theorem of calculus as one idea, what happens inside equals what crosses the boundary

L09Coordinates That Fit the Problem

CoversCurvilinear coordinates · scale factors · the line element · grad · div · curl · the Laplacian in cylindrical and spherical coordinates · why the Laplacian looks the way it does · conic sections in polar form

L10Functions Are Vectors

CoversFunction spaces · inner products · orthogonality · projection · Gram-Schmidt · symmetric (Hermitian) operators and why their eigenfunctions are orthogonal

L11Fourier Series: Every Shape Is a Chord

CoversSines and cosines as an orthogonal basis · coefficient extraction as a dot product · the complex form · convergence · Gibbs

L12The Fourier Transform

CoversThe period going to infinity · the transform pair · the uncertainty relation · convolution · the bell curve that transforms into a bell (2026-10-10, not "its own transform" under the series convention, it returns √(2π) taller)

L13Linear Differential Equations

CoversFirst-order equations and the three doors · the characteristic equation · damping regimes · driven systems and resonance · superposition · the Wronskian

L14Series Solutions and Special Functions

CoversFrobenius · the indicial equation · why Legendre · Bessel · Hermite are inevitable · generating functions · recurrence · termination as the origin of quantization

L15Partial Differential Equations: Separating the Variables

CoversThe three great equations (heat, wave, Laplace) · separation · boundary conditions as the physics · Sturm-Liouville and the orthogonality that makes it work

L16The Sphere and the Drum

CoversLaplace in spherical coordinates · spherical harmonics · the drumhead and Bessel modes · geometry alone fixing the shapes

L17The Delta Function and Green's Functions

CoversThe delta as a limit and as a rule · its transform · response to a kick · Green's functions for linear operators · superposition of kicks

L18Analytic Functions and Conformal Maps

CoversDifferentiability in the plane · Cauchy-Riemann · harmonic functions · conformal maps · flows and fields as the real and imaginary parts of one function

L19Contour Integrals and Residues

CoversCauchy's theorem · deformation of contours · Laurent series · poles and residues · the residue theorem

L20Residues at Work

CoversJordan's lemma · principal values · branch cuts · inverse transforms by contour · causality from where the contour closes

L21The Calculus of Variations

CoversFunctionals · the Euler-Lagrange equation · first integrals · constraints · least action as the organizing principle of physics

L22Tensors and the Index Notation

CoversIndex notation · Einstein summation · the Kronecker delta and Levi-Civita symbol · transformation laws · the inertia and stress tensors · the metric

L23Probability for Physicists

CoversDistributions · moments · the binomial to Poisson to Gaussian chain · random walks · the central limit theorem · error propagation

L24One Problem, Every Tool: The Drum

CoversThe whole series on one problem · polar coordinates and the Laplacian · separation of variables · Bessel functions · Fourier-Bessel expansion · Green's functions · residues · a variational bound · the noise floor