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Suppose $q(t)=A/(t-c)+r(t)$, where $r$ is continuous, the pole $c$ is fixed, and differentiable endpoints obey $a(x)<c<b(x)$ locally. ThenApply the chain rule and the fundamental theorem of calculus to getThe same boundary expression as in the Leibniz integral rule thus applies to this explicitly specified Cauchy principal value. If $A\ne 0$, the two ordinary one-sided improper integrals diverge, so their sum is not an ordinary integral. A finite boundary derivative alone does not make that divergent integral meaningful.

Among curves of fixed arc length $L>2a$ joining $(-a,0)$ to $(a,0)$, the hanging catenary globally minimizes $\int yds$. Parametrize a competitor by arc length $s\in [-L/2,L/2]$, so $x^{\prime 2}+y^{\prime 2}=1$. Integration by parts and the fixed horizontal endpoint separation give, for every $b>0$,The inequality is the Cauchy-Schwarz inequality. Equality requires $x^{\prime }=b/\sqrt{b^2+s^2}$ and $y^{\prime }=s/\sqrt{b^2+s^2}$. Integration gives the catenaryThe last equation has a unique positive $b$, since $\sinh q/q$ increases strictly for $q>0$. The equality curve is a single-valued graph below the chord, so the same global bound proves the result under those additional restrictions.

Let $\mathbf{p}\ne 0$ and choose an auxiliary vector $\mathbf{q}$ with $\mathbf{p}\times \mathbf{q}\ne 0$. Its component perpendicular to $\mathbf{p}$ is ${\mathbf{q}}_{\perp }=\mathbf{q}-(\mathbf{p}\cdot \mathbf{q})\mathbf{p}/|\mathbf{p}{|}^2$. Then all solutions of the Cartesian equation of a plane $\mathbf{x}\cdot \mathbf{p}=k$ areThe two free directions are nonzero, perpendicular to each other and perpendicular to $\mathbf{p}$, so they span its two-dimensional orthogonal complement. This proves completeness as well as direct substitution. If $\mathbf{p}\cdot \mathbf{q}=0$, then ${\mathbf{q}}_{\perp }=\mathbf{q}$. If the dot product is nonzero, another complete form is $\mathbf{x}=\alpha \mathbf{p}+[k-\alpha |\mathbf{p}{|}^2]\mathbf{q}/(\mathbf{p}\cdot \mathbf{q})+\beta (\mathbf{p}\times \mathbf{q})$.

The point assigned all-zero coordinates in a chosen Cartesian coordinate system. Choosing an origin identifies position displacements with vectors; a bare affine space has no distinguished origin.

For an axisymmetric magnetic vector potential, the conserved angular canonical momentum is $p_{\theta }=\gamma m{\rho }^2\dot{\theta }+q\rho A_{\theta }$. A particle arriving parallel to a solenoid axis from a field-free region has $p_{\theta }=0$. In a uniform field, $A_{\theta }=B\rho /2$ and $\dot{\theta }=-qB/(2\gamma m)$ wherever $\rho \ne 0$. The mechanical angular momentum changes through the fringe while the canonical quantity remains constant. Static magnetic fields also conserve $\gamma m{c}^2$; the azimuthal kick therefore reduces axial kinetic motion rather than increasing total speed.

On a common invariant domain for the operator products, let the position operator and momentum operator obey $[\hat{x},\hat{p}]=i\hslash I$. Then $\hat{x}\hat{p}=\hat{p}\hat{x}+i\hslash I$, so both displayed expressions equal $(\hat{p}\hat{x}{)}^2+i\hslash \hat{p}\hat{x}$. In position representation both act on a smooth function $f$ as $-{\hslash }^2(x^2{f}^{\prime \prime }+2x{f}^{\prime })$. This particular equality does not imply that arbitrary classical reorderings remain equivalent in quantum mechanics.

Putting $M=m$ in the diatomic-chain dispersion relation with equal springs gives ${\omega }_{-}=2\sqrt{K/m}|\sin (kd/2)|$ and ${\omega }_{+}=2\sqrt{K/m}|\cos (kd/2)|$ in the reduced zone $|k|\le \pi /(2d)$. The branches touch at its edge. If the sites are actually identical, the true primitive unit cell has length $d$; the apparent optical phonon branch is the outer part of the monatomic-chain dispersion relation folded by an unnecessarily doubled cell. A gap does not appear merely from redrawing the zone.

Measure $\theta \in [-\pi /2,\pi /2]$ from the vertically downward direction on the lower half of a cylinder of radius $R$, and let $z$ be axial distance. From rest at an edge, conservation of energy gives speed $\sqrt{2gR\cos \theta }$. The travel time is $T=(2gR{)}^{-1/2}\int \sqrt{R^2+z^{\prime 2}}/\sqrt{\cos \theta }d\theta$. Since $z$ is a cyclic coordinate, its conjugate quantity $z^{\prime }/[\sqrt{\cos \theta }\sqrt{R^2+z^{\prime 2}}]=K$ is constant, giving the displayed formula. For finite positive axial separation, $0<K<1$ is fixed uniquely by integrating $z^{\prime }$ between the edges: the separation grows strictly from zero to infinity as $K$ increases from zero to one. The time integrand is strictly convex in $z^{\prime }$, so its supporting-line inequality proves global minimality for the fixed endpoints. As the starting edge is approached, $z^{\prime }\to 0$: the initial direction is directly down the curved surface, with no axial component. This differs from the constant-pitch shortest geodesic on the unrolled cylinder.

For a real scalar lattice-periodic potential, let $|n⟩$ be a nondegenerate zone-centre periodic eigenstate with energy $E_n$. Conjugation by $e^{i\mathbf{k}\cdot \mathbf{r}}$ gives $H(\mathbf{k})=H(0)+\hslash \mathbf{k}\cdot \mathbf{p}/m+{\hslash }^2{k}^2/(2m)$. The zone-centre state can be chosen real, so $⟨n|\mathbf{p}|n⟩=0$. Second-order nondegenerate perturbation theory then givesTime-reversal symmetry makes this isolated spinless band even in $\mathbf{k}$. In a direction $\mathbf{e}$, the curvature givesA degeneracy requires a matrix treatment before extracting curvatures; a nonlocal potential or spin-orbit terms can also alter the simple momentum vertex.

A blackbody spectrum is the spectrum of ideal black-body radiation in thermal equilibrium, given by Planck's law. Thus Planck spectrum and blackbody spectrum name the same ideal spectrum.

In a thin rectangular electromagnetic cavity, the window $\hslash c\pi /L_x,\hslash c\pi /L_y\ll k_{B}T\ll \hslash c\pi /L_z$ freezes the nonzero thickness indices while retaining many in-plane normal modes. The zero-thickness-index family has one polarization of an electromagnetic wave and density of states $dN=Akdk/(2\pi )=2\pi A\nu d\nu /c^2$. Multiplying by $h\nu /(e^{h\nu /k_{B}T}-1)$ and integrating gives $U_{2D}=4\pi \zeta (3)A(k_{B}T{)}^3/(h^2{c}^2)$. Thus thermal energy scales as area times $T^3$. The Riemann zeta function appears through ${\int }_0^{\infty }{x}^2/(e^x-1)dx=2\zeta (3)$.

This is an intermediate low-temperature regime of an anisotropic finite cavity. When $k_{B}T$ also falls below the first in-plane excitation, thermal energy becomes exponentially small. If a different model excludes every zero thickness index, its first transverse mode has a finite gap and the proposed $T^3$ regime does not occur. In the three-dimensional continuum regime the corresponding density of states is proportional to volume times ${\nu }^2$ and thermal energy is proportional to volume times $T^4$.

The steady electric potential differences and electric currents about which signals vary in an electrical circuit. A small-signal model approximates nearby behavior.

For a p-n junction, biasing with the p-type semiconductor at negative electric potential difference relative to the n-type semiconductor.

For a p-n junction, biasing with the p-type semiconductor at positive electric potential difference relative to the n-type semiconductor.

The valley of stability is the region of nuclear masses near the minima with respect to proton-neutron imbalance. Its beta-stable band bends toward neutron excess for heavier nuclei, as nuclear Coulomb energy competes with the cost of neutron-proton asymmetry. The band is distinct from the particle-emission boundaries set by nuclear separation energies.

The spin-independent vector contribution to an allowed beta decay. The isospin operator ${\tau }_k^{\pm }$ changes a neutron into a proton or the reverse; the sign convention depends on the assigned nucleon isospin projection. Its rotational rank is zero, so $J_f=J_i$ and the parity is unchanged. The phrase Fermi-type forbidden contribution extends the $S=0$ classification to higher orbital angular momentum.

The spin-dependent axial-vector contribution to an allowed beta decay. Its rotational rank is one, permitting $\Delta J=0,1$ with unchanged parity, but excluding $0\to 0$. Here ${\mathbf{\sigma }}_k$ acts on nucleon spin and ${\tau }_k^{\pm }$ on isospin. Higher-orbital spin-dependent contributions are conventionally described as Gamow-Teller-type forbidden transitions.

A beta decay transition whose leading lepton orbital term has $l=0$. Fermi transitions allow $\Delta J=0$ and Gamow-Teller transitions allow $\Delta J=0,1$, excluding a Gamow-Teller $0\to 0$ transition. Both preserve nuclear parity. Allowed describes the selection rules; cancellations in the nuclear matrix element can still make an allowed transition slow.