The causal magnetic vector potential generated by a current density in the Lorenz gauge, with no additional incoming homogeneous field. Each source contribution is evaluated at its retarded time. The integral is a convolution with the retarded wave-equation Green function; its validity includes the stated boundary and source conditions.

At normal incidence, an ideal grating of $N$ lines and period $d$ resolves a small wavelength separation $\delta \lambda$ in order $m$ by the Rayleigh condition $|m|N\gtrsim \lambda /\delta \lambda$. Combining this with $|m|\lambda =d|\sin \theta |$ yields $W=Nd\gtrsim {\lambda }^2/(\delta \lambda |\sin \theta |)\ge {\lambda }^2/\delta \lambda$. The final expression is a limiting lower bound as diffraction approaches grazing. Finite usable diffraction angle, unequal line strengths, aberrations and intrinsic linewidth can all demand a larger instrument or a more detailed criterion.

A photon rocket converts stored mass-energy into a collimated beam of photons. In its instantaneous rest frame, energy conservation gives $c^{2}dM=-d{E}_{\gamma }$, and momentum conservation gives $Mcd\eta =d{E}_{\gamma }/c$. Therefore a monotone change in rapidity costs $M_f/M_i=e^{-|\Delta \eta |}$. The rocket's rest mass includes its fuel and stored internal energy; inefficiency or an angular spread of the exhaust increases the fuel requirement.

For an isolated decay of a parent of rest mass $M$ into products of rest masses $m_i$, the difference between total final and initial kinetic energies is $(M-{\sum }_i{m}_i)c^2$ in every inertial frame. Conservation of total energy and inclusion of all products are essential.

For $T^{\mu \nu }=\rho u^{\mu }{u}^{\nu }$ and constant normalized $u_{\mu }{u}^{\mu }=c^2$, contraction of ${\nabla }_{\mu }{T}^{\mu \nu }=0$ with $u_{\nu }$ gives ${\nabla }_{\mu }(\rho u^{\mu })=0$. At nonzero density the remaining projection gives $u^{\mu }{\nabla }_{\mu }{u}^{\nu }=0$.

Compare two observers at one event using a common static orthonormal frame outside a Schwarzschild horizon. If one observer moves tangentially at speed $u$ and another radially at speed $w$, their four-velocity contraction is ${\gamma }_u{\gamma }_w{c}^2$. Their relative speed therefore has the displayed form. The radial component measured by the tangential observer is $w/{\gamma }_u$, so Euclidean subtraction of the two original three-velocities fails. For a circular timelike geodesic and radial fall from rest at infinity at radius $R$, the relative speed is $c\sqrt{3\mu /R}$, where $\mu =GM/c^2$ and $R>3\mu$.

A regular singular point with a full set of analytic local solutions is an apparent singularity of the differential equation. Singular normalized coefficients do not by themselves force singular solutions. For example, $x{y}^{\prime \prime }-y^{\prime }=0$ has normalized coefficient $-1/x$, but its fundamental solutions $1,x^2$ are both analytic through zero. Their Wronskian is $2x$, which vanishes at the singular point while being nonzero on each adjacent ordinary interval. The nonvanishing-Wronskian criterion for a fundamental set applies on intervals where the normalized equation has regular coefficients.

A silver electrode in contact with silver sulfate and a solution of controlled sulfate activity. The reduction is $\mathrm{A}{\mathrm{g}}_2\mathrm{S}{\mathrm{O}}_4+2{\mathrm{e}}^{-}\rightleftharpoons 2Ag+\mathrm{S}{\mathrm{O}}_4^{2-}$.

The conventional zero of standard electrode potentials, based on hydrogen oxidation/reduction at specified standard hydrogen fugacity and proton activity.

A real triplet with a rotationally invariant common mass term has three conserved internal currents. With the displayed sign convention, the current for $\delta \mathbf{\varphi }=\theta \mathbf{n}\times \mathbf{\varphi }$ is the negative of the conventional canonical current contracted with $\mathbf{n}$. Overall current sign does not affect conservation. Integrated charges require convergent integrals and vanishing net boundary flux; a local continuity equation alone does not guarantee conservation in an open region.

For reaction Jacobian $J=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ with $\mathrm{tr}J<0$ and $\det J>0$, a Fourier mode evolves under $J-q^2\mathrm{diag}(D_u,D_v)$. Its determinant is $D_u{D}_v{q}^4-(D_{v}a+D_{u}d)q^2+\det J$. A finite unstable band requires $D_{v}a+D_{u}d>0$ and $(D_{v}a+D_{u}d{)}^2>4D_u{D}_v\det J$, with an allowed domain mode in that band.

A trial function is a chosen candidate used to approximate a solution or evaluate a variational principle. For an eigenvalue upper bound from a Rayleigh quotient, it must be nonzero, have finite numerator and denominator, and lie in the appropriate variational domain. In a Dirichlet problem it satisfies the fixed endpoint values. Natural boundary conditions need not always be imposed on a trial in the quadratic-form domain, but a candidate violating an essential boundary condition does not give the stated bound.

For real smooth coefficients with $p,w>0$, the Rayleigh quotient $\Lambda [y]=\int [p(y^{\prime }{)}^2+q{y}^2]dx/\int w{y}^{2}dx$ has the displayed first variation. Its stationary points solve the Sturm-Liouville equation on an admissible class whose boundary variations make $[p{y}^{\prime }\eta {]}_a^b=0$. Homogeneous Dirichlet boundary conditions do this by setting $\eta =0$ at the ends; homogeneous Neumann boundary conditions do it through $y^{\prime }=0$. More general self-adjoint boundary conditions can require a boundary contribution to the quadratic form.

For an eigenfunction, the weaker identity $[py{y}^{\prime }{]}_a^b=0$ suffices to give $\lambda =\Lambda [y]$ by integration by parts, but it does not by itself imply the condition on every variation. An explicit counterexample uses $p=q=w=1$ on $[0,1]$ and $y=\cos \pi x+\sin \pi x$. This solves $-y^{\prime \prime }+y=(1+{\pi }^2)y$ and has $[y{y}^{\prime }{]}_0^1=0$. Define a nearby familyIt keeps $y_{\epsilon }(0)=1$, $y_{\epsilon }^{\prime }(0)=\pi$, $y_{\epsilon }(1)=-1+\epsilon$, and $y_{\epsilon }^{\prime }(1)=\pi /(-1+\epsilon )$, so $[y_{\epsilon }{y}_{\epsilon }^{\prime }{]}_0^1=0$ for small $\epsilon$. Nevertheless, $\eta ={\partial }_{\epsilon }{y}_{\epsilon }{|}_0$ has $\eta (0)=0$, $\eta (1)=1$, and ${\int }_0^1{y}^{2}dx=1$, giving $\delta \Lambda =2[y^{\prime }\eta {]}_0^1=-2\pi \ne 0$. Boundary-energy cancellation alone therefore cannot replace a specified variational domain.

For a nonzero trial function with $y(0)=y(L)=0$ and $y(L-x)=y(x)$, substituting $L-x$ for $x$ proves ${\int }_0^{L}x{y}^{2}dx=(L/2){\int }_0^L{y}^{2}dx$. This gives the displayed Rayleigh quotient. The Poincaré inequality on an interval bounds it below by $2{\pi }^2/L^3$, with equality for a nonzero multiple of $\sin (\pi x/L)$. Thus this sine is the best reflection-symmetric trial for $-y^{\prime \prime }=\lambda xy$, while the unrestricted lowest eigenfunction need not be reflection-symmetric. The polynomial $x(L-x)$ gives $20/L^3$, only a factor $10/{\pi }^2$ above the sine's value. Both are upper bounds on the unrestricted lowest eigenvalue by the Rayleigh-Ritz method.

For a self-adjoint operator with two normalized orthogonal eigenvectors $u_0,u_1$ and $E_0<E_1$, the Rayleigh quotient of ${\psi }_t=u_1+t{u}_0$ is $(E_1+t^2{E}_0)/(1+t^2)$. It is stationary at $t=0$ but has second derivative $2(E_0-E_1)<0$ there. Thus an exact excited eigenfunction can be a maximum within a trial family. A constrained minimum requires exclusion of lower states, as in an excited-state variational constraint.

For fixed temperatures and stress-free impermeable boundaries, linear marginal Rayleigh-Bénard convection with dimensionless horizontal wavenumber $q$ obeys $\mathrm{Ra}(q)=(q^2+{\pi }^2{)}^3/q^2$. Differentiation gives $q_c=\pi /\sqrt{2}$ and the displayed minimum. The threshold depends on the mechanical boundary conditions: two rigid no-slip boundaries instead give approximately $1708$.

Here isotropic means rotationally invariant in three dimensions, a stronger condition than merely having identity covariance matrix. Such a random vector has uniform direction on the sphere, independent of its nonzero length $R$. Its projection onto a fixed plane has length $R_{\perp }=R\sin \theta$. The solid-angle averages areThus $\mathbb{E}{R}_{\perp }=(\pi /4)\mathbb{E}R$ and $\mathbb{E}{R}_{\perp }^2=(2/3)\mathbb{E}{R}^2$ whenever the corresponding moments exist. A mass at zero contributes zero to both lengths and needs no direction. The factors are independent of the radial law; the mean-length factor differs from the root-mean-square factor.

Raman spectroscopy enhanced by a nearby structured surface, commonly through local plasmonic fields and adsorbate-specific electronic interactions. Enhancement of a signal per molecule does not alone determine a detection limit.

An ideal receding mirror reflecting a source beam of constant power $P$ receives force $(2P/c)(1-\beta )/(1+\beta )$ after including both Doppler-shifted reflected momentum and reduced impact rate. For fixed rest mass $M$, use $dt/d\tau =\gamma$ in the momentum equation. It reduces to $d\beta /d\tau =(1-\beta {)}^2/{\tau }_0$, where ${\tau }_0=M{c}^2/(2P)$. Starting from rest gives $\beta =\tau /(\tau +{\tau }_0)$. This continuum mirror model neglects individual-photon recoil and finite beam interception.

A region that concentrates solids by opposing their loss through radial drift. Near a radial gas-pressure maximum, the pressure-support drift changes sign and carries particles toward the maximum from both sides. Gas advection and turbulent diffusion can limit retention.