Foundations

A reusable measurement report identifies the measurand, value and unit, relevant conditions and corrections, and whether uncertainty is standard u_c or expanded U.

An illustrative temperature model y=x̄+C+δ has x̄=20.00 °C from four independent readings with s=0.20 K, C=+0.50 K with u_C=0.20 K, and independent zero-centered δ uniform on ±0.05 K.

An instrument calibration curve models indication y as a function of reference input x, such as y=a+bx+ε.

For the valid linear calibration y=a+bx, an unknown input is estimated as x̂=(y₀−â)/b̂.

Calibration-model validation examines residuals for curvature, unequal variance, dependence and outliers, and compares independent or check-standard results across the working range.

A measurement standard realizes a specified quantity with a stated value and uncertainty for reference use.

A reference material is sufficiently homogeneous and stable for specified properties and an intended measurement or examination use.

A detection limit depends on the measurement procedure's false-presence probability α and false-absence probability β.

Results for quantities of the same kind are metrologically comparable when traceable to the same reference.

An ideal baseband signal containing no frequencies above B can be reconstructed from exact uniform samples at f_s>2B.

For a uniform rounding quantizer with step Δ and no overload, the error magnitude is at most Δ/2.

For repeated indications, the mean has variance Var(x̄)=(1/n²)Σ_iΣ_j Cov(x_i,x_j).

Instrumental drift is a continuous or incremental change in indication caused by changes in an instrument's metrological properties.

Fixing a defining constant's numerical value makes that SI definition exact; it does not make an experiment or transferred standard exact.

A primary caesium frequency standard relates its oscillator to the defining unperturbed Cs-133 hyperfine transition by estimating and correcting design-dependent shifts, including atomic-motion Doppler and thermal blackbody effects.

With calibrated light frequency f, vacuum wavelength is λ₀=c/f.

A Kibble balance combines weighing mg=I Bℓ with velocity-mode U=vBℓ to eliminate the common geometry-field factor: mgv=IU.

For a conservative central interaction U(r), relative motion is planar and angular momentum L=μr²θ̇ is conserved.

For a frame rotating at constant Ω about a common inertial origin, a_inertial=a_rot+2Ω×v_rot+Ω×(Ω×r).

For ideal independent holonomic constraints f_α(q,t)=0, constrained Lagrange equations can retain redundant coordinates through d(∂L/∂q̇_i)/dt−∂L/∂q_i=Σ_α λ_α∂_i f_α.

In canonical coordinates, {A,B}=Σ_i[(∂_q_iA)(∂_p_iB)−(∂_p_iA)(∂_q_iB)] and dA/dt={A,H}+∂_tA.

A canonical transformation changes phase-space coordinates and the Hamiltonian while preserving Hamilton's description of the same physical trajectories.

A thin spherical lens in a common surrounding medium has 1/f=(n_lens/n_medium−1)(1/R₁−1/R₂).

For scalar co-polarized fields, averaged intensity includes a cross-correlation term.

For equal-phase coherent illumination of narrow slits separated by d, far-field bright directions satisfy d sinθ=mλ and dark directions d sinθ=(m+½)λ.

A monochromatic scalar field in a uniform medium propagates paraxially as E_z(x,y)≈e^(ikz)/(iλz)∫∫E₀(ξ,η)exp[ik((x−ξ)²+(y−η)²)/(2z)]dξdη, with k=2π/λ.

When the input-aperture quadratic phase in the Fresnel integral is negligible, the far-field amplitude is proportional to the two-dimensional Fourier transform of the aperture field evaluated at spatial frequencies x/(λz), y/(λz).

At normal incidence on a periodic grating of pitch d, principal propagating orders satisfy d sinθ_m=mλ.

A uniform ideal circular pupil of diameter D gives an Airy intensity pattern whose first dark ring has small angle θ≈1.22λ/D.

For a linear shift-invariant scalar imaging system, coherent image amplitude is E_image=E_object*h.