For $A,h>0$, a rectangular cross section of width $w$ and height $h$, capped by a symmetric triangular roof at angle $0<\theta <\pi /2$, has area $A=wh+w^2\tan \theta /4$. Its total sloping roof length is $w\sec \theta$. A cost $2\alpha h+\beta w\sec \theta$ with $\beta >0$ has the same minimizing shape as this roof length, since the fixed wall term is constant.

The Lagrange multiplier equations for $w\sec \theta$ under the area constraint give $w(1-{\tan }^2\theta )=4h\tan \theta$. This is the first displayed condition; it also implies $\theta <\pi /4$. Eliminating $\tan \theta =4(A-wh)/w^2$ gives the second condition.

For a global check, the squared roof length is $Q(w)=w^2+16(A-wh{)}^2/w^2$ for $0<w<A/h$. Its derivative has the sign of $w^4+16Ahw-16A^2$, a strictly increasing expression with one root in that interval. Thus this root is the unique minimum. The shape is independent of the positive roof-cost coefficient and the constant wall cost. If $\beta =0$, every feasible shape has the same cost and these conditions are no longer necessary.

For a lidless rectangular parallelepiped with positive dimensions $l,w,h$ and prescribed external surface area $S=lw+2lh+2wh$, the largest volume is attained at $l=w=2h=\sqrt{S/3}$. The arithmetic-geometric mean inequality applied to $lw,2lh,2wh$ giveswith equality exactly when these three face-area terms are equal. This proves both the claimed global bound and its attainable equality case. Lagrange multipliers produce the same dimensions from the interior stationarity equations. The weighted open-box minimization is the corresponding fixed-volume formulation with general positive face weights.

A speed distribution is the probability distribution of the magnitude of velocity. When it has a probability density function $f$, the fraction of particles with speeds between $v$ and $v+dv$ is $f(v)dv$, and ${\int }_0^{\infty }f(v)dv=1$. The Maxwell speed distribution describes an equilibrium ideal gas; the effusive beam speed distribution weights that population by its rate of arrival.

Thermal motion is microscopic random motion associated with a system's temperature; Brownian motion makes its effects on larger suspended grains visible.

In a dilute gas whose mean free path $\lambda$ is short compared with macroscopic gradient lengths, particles carry properties from their previous collisions over distances of order $\lambda$. A first-gradient isotropic estimate gives a transported flux $-n\overline{v}\lambda \nabla a/3$ for a property $a$ carried per particle. Applying this reasoning to species abundance, directed momentum $m{u}_x$ and thermal energy with per-particle heat capacity $c_v$ gives diffusivity $D\sim \overline{v}\lambda /3$, dynamic viscosity $\eta \sim nm\overline{v}\lambda /3$ and thermal conductivity $\kappa \sim n{c}_v\overline{v}\lambda /3$. These numerical prefactors are estimates, not exact collision-integral results.

For a constant hard-sphere diameter $d$, $\lambda \propto 1/(n{d}^2)$ and $\overline{v}\propto \sqrt{T/m}$. Consequently $\eta \propto \sqrt{mT}/d^2$, $\kappa \propto c_v\sqrt{T/m}/d^2$ and $D\propto \sqrt{T/m}/(n{d}^2)$. At fixed pressure the last scales as $T^{3/2}/p$. The near pressure-independence of viscosity and thermal conductivity applies only while gas-gas collisions, rather than apparatus walls, determine the mean free path. Temperature-dependent cross-sections and internal molecular excitations change the simple power laws.