A pressure measurement determines pressure relative to a specified reference using an instrument such as a manometer or pressure transducer. Gauge pressure, absolute pressure and differential pressure require different references; measurement calibration and measurement uncertainty determine how the reading can be interpreted.

The time for a precessing orientation to turn through $2\pi$. An unlabelled line is unchanged by a $\pi$ turn, so its visible pattern can repeat after half the full precession period.

A region of lower potential energy surrounded by higher potential energy; a particle must gain sufficient energy to escape classically.

A quadratic potential energy about a stable equilibrium. In one dimension $k>0$ is the restoring stiffness and the force is $-dV/dx=-kx$. A mass $m$ undergoing undamped motion has angular frequency $\sqrt{k/m}$. A smooth potential with a nondegenerate local minimum is approximately harmonic for sufficiently small displacements; higher-order terms describe departures from simple harmonic motion.

A polymer globule is a compact conformation favored by effective segment attractions in a poor solvent. If its internal segment density is independent of chain length, its volume is proportional to $N$ and a size measure scales as $N^{1/3}$. The proportionality between physical volume and the cube of a radius of gyration depends on shape and density profile. Compactness alone supplies scaling, not a uniquely determined numerical monomer volume.

For a symmetric self-gravitating slab with polytropic equation of state $p=K{\rho }^2$ and constant $K>0$, hydrostatic equilibrium gives $2K{\rho }^{\prime }=-{\psi }^{\prime }$. The plane-parallel Poisson equation for Newtonian gravity then yields ${\rho }^{\prime \prime }+(2\pi G/K)\rho =0$. Reflection symmetry at the midplane fixes the title profile. For a vacuum surface, retain the nonnegative profile only on $|z|\le z_s=\pi /(2k)$ and put $\rho =0$ outside. The half-thickness is independent of central density in this ideal index-one model, while the surface mass density is $\Sigma =2{\rho }_c/k$.

A polynomial factor of each of two specified polynomials. A nonconstant common factor prevents them from being coprime polynomials.

An octahedron is a polyhedron with eight faces. The regular octahedron has six vertices, twelve equal edges and eight equilateral triangular faces. One realization has vertices $\pm {\mathbf{e}}_1,\pm {\mathbf{e}}_2,\pm {\mathbf{e}}_3$; two vertices are adjacent exactly when they are not an opposite pair. Its edge graph is the octahedral graph.

A pyramid is a polyhedron with a polygon as its base and triangular faces joining the base edges to one apex outside the base plane. If its base area is $A$ and perpendicular height is $h$, its volume is $Ah/3$. At height $z$ above the base, similarity gives cross-sectional area $A(1-z/h{)}^2$. Integrating this area from $0$ to $h$ proves the formula.

At an edge of a convex polyhedron, its interior dihedral angle is the angle of the wedge inside the solid, measured in a plane perpendicular to the edge. If the two incident faces have outward unit normal vectors ${\hat{\mathbf{n}}}_1,{\hat{\mathbf{n}}}_2$, then their normal angle is supplementary to the interior angle. Thus $\cos \theta =-{\hat{\mathbf{n}}}_1\cdot {\hat{\mathbf{n}}}_2$. Equivalently, subtract from each vector pointing into one face its orthogonal projection onto the common edge; the angle between the resulting inward face directions is the interior angle. This distinguishes an interior angle from the acute angle convention for two unoriented planes.

A polygon with six sides and six vertices. A regular hexagon has equal sides and angles, with each interior angle equal to $2\pi /3$. Its symmetry group consists of six rotations and six reflections, forming a dihedral group of order twelve.

A polygon with five sides and five vertices. A regular pentagon has equal sides and angles; its symmetry group is the order-ten dihedral group, consisting of five rotations and five reflections. Each reflection axis passes through a vertex and the midpoint of the opposite edge.

A sphere with constant electric polarization $\mathbf{P}=P\hat{\mathbf{z}}$ has zero volume bound charge and surface bound charge density ${\sigma }_b=P\cos \theta$. Integrating $\mathbf{r}{\sigma }_{b}dA$ gives the displayed electric dipole moment. The surface distribution is purely degree one in spherical harmonics, so its external static electric potential is exactly dipolar. Its interior field is finite and is not that of a point singularity.

The electric field produced by the bound charge of a polarized body is its depolarization field. For a uniformly polarized ellipsoid in vacuum, the internal depolarization field is uniform and opposes the relevant component of the electric polarization: $E_{\mathrm{dep},i}=-N_i{P}_i/{ϵ}_0$. Its dimensionless geometric factors obey $N_x+N_y+N_z=1$.

A Poisson kernel represents the harmonic extension of Dirichlet boundary data by an integral over the boundary. When the Dirichlet Green function is normalized by $-\Delta G_{\Omega }=\delta$, its negative outward normal derivative gives the displayed kernel. The Poisson kernel for the upper half-space and the Poisson kernel for the upper half-plane give explicit examples with different boundary dimensions. Boundedness or other stated growth conditions are needed when selecting a solution in an unbounded domain.

For $\beta >0$, solve ${\nabla }^2\varphi =-\rho$ in three dimensions with $\rho (r)=A{e}^{-\beta r}/r$, finite potential at the origin and decay at infinity. The Green function is $1/(4\pi |\mathbf{r}-{\mathbf{r}}^{\prime }|)$ in this sign convention. The angular integral supplied by the spherical shell theorem givesThe value at the origin is the limit $A/\beta$. A second derivation sets $u=r\varphi$: the radial Poisson equation becomes $u^{\prime \prime }=-A{e}^{-\beta r}$. Decay excludes the constant-potential term and boundedness at the origin cancels the remaining $1/r$ singularity. Although the source is locally integrable, it is singular at the origin; the resulting potential solves the equation there in the sense of distributions, without an extra Dirac delta.

For a planar surface contacting a symmetric monovalent ideal electrolyte on one side, the Poisson-Boltzmann equation is ${\psi }^{\prime \prime }=(2e{n}_{\infty }/\epsilon )\sinh (e\psi /(k_{B}T))$. Multiplication by ${\psi }^{\prime }$ and integration using the zero field and zero potential in the bulk givesThe solution decays towards the bulk, and the boundary condition $\sigma =-\epsilon {\psi }^{\prime }(0^{+})$ therefore yields the displayed charge-potential relation, with ${\kappa }^2=2e^2{n}_{\infty }/(\epsilon k_{B}T)$. Its small-potential limit is the planar Debye-Hückel double layer. It is nonlinear but still a mean-field model with point-like ions and constant permittivity; it does not include a compact interfacial layer or finite-ion-size saturation.

For a differentiable path in plane polar coordinates with $r>0$, the moving unit vectors are $\hat{\mathbf{r}}=(\cos \theta ,\sin \theta )$ and $\hat{\mathbf{\theta }}=(-\sin \theta ,\cos \theta )$. Differentiating the position $\mathbf{r}=r\hat{\mathbf{r}}$ and using $\dot{\hat{\mathbf{r}}}=\dot{\theta }\hat{\mathbf{\theta }}$ proves the displayed velocity. Its signed radial and transverse vector components are $\dot{r}$ and $r\dot{\theta }$. A second differentiation gives the acceleration in polar coordinates. The complex representatives of the two moving unit vectors are $e^{i\theta }$ and $i{e}^{i\theta }$.

For a source-free plane electromagnetic wave in vacuum, let $n^{\mu }=(1,\mathbf{s})$ be its null propagation Lorentz four-vector. In signature $(+---)$, take $F^{0i}=-E_i/c$, $F^{ij}=-{ϵ}_{ijk}{B}_k$, and define the dual electromagnetic field tensor by ${\tilde{F}}^{0i}=-B_i$, ${\tilde{F}}^{ij}={ϵ}_{ijk}{E}_k/c$. Lower both indices with the Minkowski metric. Direct contraction givesBoth contractions vanish exactly when the transverse plane electromagnetic wave Maxwell equations hold. Being zero covectors, they remain zero in every inertial frame. Thus the transformed spatial wavevector is perpendicular to both transformed fields, even though the separate three-dimensional dot products are not themselves Lorentz scalars for arbitrary fields.

One contraction alone is insufficient without the other Maxwell equations. In particular, adding a magnetic field parallel to $\mathbf{s}$ leaves $n^{\mu }{F}_{\mu \nu }$ unchanged but violates $\mathbf{s}\cdot \mathbf{B}=0$. Conversely the two scalar perpendicularity conditions alone do not impose the required relation $\mathbf{B}=\mathbf{s}\times \mathbf{E}/c$ between field strengths and directions.

An involute is the curve traced by the end of a taut string as it is unwound from a fixed curve. For a circle of radius $R$, one parametrization is $\mathbf{x}(\theta )=R(\cos \theta +\theta \sin \theta ,\sin \theta -\theta \cos \theta )$. The straight free length is $R\theta$, and differentiation shows that the traced velocity is perpendicular to the taut string. Reversing the motion winds the string back around the circle.