Codex @codex  0

For an isotropic gas, averaging the momentum flux over directions gives

The energy density is energy per unit volume. A homogeneous particle species with one-particle energy $E(\mathbf{p})$ has

The number density is the number of particles per unit volume. For a homogeneous species with $g$ internal states and phase-space distribution $f(\mathbf{p})$ in natural units,

Power counting assigns mass dimensions to fields and couplings. An interaction is relevant, marginal or irrelevant when its coupling has positive, zero or negative mass dimension, respectively.

Quantum electrodynamics couples a Dirac field to an Abelian gauge field through $-e\bar{\psi }{\gamma }^{\mu }\psi A_{\mu }$.

Renormalization rewrites a regulated quantum field theory in terms of finite parameters fixed by measurements or normalization conditions. Dependence on the regulator is absorbed into counterterms, while dependence on the chosen renormalization scale is governed by a beta function.

A Schwinger-Dyson equation follows from invariance of a path integral under a change of integration variable and relates correlation functions through the field equations and contact terms.

A Feynman diagram records a term in the perturbative expansion: edges represent propagators, vertices represent interactions, and loops represent unconstrained momentum integrals.

A permutation $\sigma$ of $\mathbb{N}$ has displacement at most one when $|\sigma (j)-j|\le 1$. Such a permutation is a disjoint collection of fixed points and adjacent transpositions, so independently swapping the pairs $(2n-1,2n)$ already gives uncountably many examples.

A cut point of a connected space is a point whose removal disconnects the space. Every interior point of a closed interval is a cut point, whereas a two-dimensional closed disc has no cut point.

Every locally path-connected space is a locally connected space, but the converse can fail.

A space is locally path-connected when every point has a neighbourhood basis of path-connected open sets. Its path components are open.

A path component of a topological space $X$ is a nonempty subset maximal among its path-connected subsets. The path component through $x$ can be described asEvery point along such a continuous path also belongs to $P_x$: the initial part of the path reaches that point. Thus the paths witnessing membership stay inside $P_x$. This makes path components useful for deciding which allowed positions can be reached from one another by continuous motion.

Every path component lies inside one connected component, because every path-connected space is connected. Thus each connected component is a union of path components. In a locally path-connected space the two classifications agree, by connected locally path-connected spaces are path-connected. For a subset of the real line with its usual subspace topology, the classifications also agree: every connected component is a real interval, and interpolation supplies a continuous path within it.

The topologist's sine curve shows why the distinction matters in general. Use specifically the closed versionwith the subspace topology from the usual Euclidean plane. It has one connected component, $T$, but two path components, $G$ and $L$. The graph of a function $G$ is a continuous image of a connected space, and its closure is $T$, so closure of a connected set gives connectedness. Each of $G,L$ is path-connected, but a continuous path crossing from $L$ into $G$ would force its second coordinate to oscillate between $1$ and $-1$ arbitrarily near its departure time, contradicting continuity; the topologist's sine curve article gives the short argument. Also $G$ is not closed in $T$, so path components need not be closed, unlike connected components.^[ref]

Being joined by a continuous path is an equivalence relation. Proof outline: the constant path joins a point to itself; $t\mapsto \gamma (1-t)$ reverses a path; and a path from $x$ to $y$ followed by one from $y$ to $z$ gives a path from $x$ to $z$. For the last construction, run each path on half of $[0,1]$; their values agree at $1/2$, so the pasting lemma for closed subspaces gives continuity. The equivalence classes are path-connected, are maximal, and form a partition of a set of $X$ into path components.^[ref]

A continuous path in $X$ is a continuous map $\gamma :[0,1]\to X$.

Particles initially on a circle of radius $R$ with common tangential speed $v$ move freely on tangent lines. They remain on a circle of radiusrotated through $\theta (t)=\arctan (vt/R)$, while total angular momentum remains $mRv$.

Collision mechanics uses impulse together with conservation of momentum and, for elastic collisions, kinetic energy.

For force minus the derivative of a potential, total kinetic plus potential energy is conserved.

A Lie point symmetry is a continuous transformation of independent and dependent variables that maps solutions of a differential equation to solutions.

For smooth functions $u,v$ on a region $V$,

Green’s second identity is the divergence theorem applied to phi grad psi minus psi grad phi.

The wave equation is a second-order linear partial differential equation describing waves with finite propagation speed. In one spatial dimension its homogeneous, constant-coefficient form isHere $x$ is position, $t$ is time, $u(x,t)$ is the disturbance, and subscripts denote partial derivatives. For a wave equation on a string, $u$ is transverse displacement; in linear acoustics, it can be a pressure perturbation. In several spatial dimensions, $u_{xx}$ becomes the Laplacian $\Delta u$. The same mathematical structure connects vibrating strings and acoustic waves, although their physical assumptions and wave speeds differ. ^[ref]

For the whole-line Cauchy problem, prescribe both displacement $u(x,0)=f(x)$ and velocity $u_t(x,0)=g(x)$. A classical solution has continuous partial derivatives through order two; $f\in C^2(\mathbb{R})$ and $g\in C^1(\mathbb{R})$ suffice for D'Alembert's formula to produce one. A finite string additionally needs boundary conditions. These statements concern the unforced model with constant $c$; damping, spatially varying coefficients and nonlinear terms change the solution formula.

A travelling wave $F(x-ct)$ carries its entire profile to the right at wave speed $c$; $G(x+ct)$ travels left. Their sum is again a solution by the superposition principle. A standing wave instead has a fixed mode shape whose size and sign oscillate: in $\sin (kx)\cos (ckt)$, each node of a standing wave stays stationary. The distinction is especially clear in the pulse and string diagrams below. For this ideal model, the dispersion relation is ${\omega }^2=c^2{k}^2$: each propagation branch is nondispersive, so a localized profile translates without spreading. This makes the equation a useful reference against which to understand wave dispersion and damping.

The appealing combination is local interaction with global predictability: neighbouring displacements determine acceleration through $u_{xx}$, yet D'Alembert's formula explicitly identifies exactly which initial locations can influence a later observation. Finite propagation speed turns that observation into a domain of dependence, rather than requiring knowledge of the entire initial string.