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For the surface of revolution of the circle with center $(a,R)$ and radius $r<R$ in the upper half-plane, use $X(u,v)=(a+r\cos u,(R+r\sin u)\cos v,(R+r\sin u)\sin v)$ and unit normal $n=(\cos u,\sin u\cos v,\sin u\sin v)$. Exactly four surface normals pass through the displayed $P$. The equation $P-X=tn$ forces $(t+r)\cos u=0$. The case $t=-r$ would put $P$ at radial distance $R$, so $\cos u=0$; its nonzero radial coordinate then forces $v=0$ or $\pi$. All four combinations work. By contrast, infinitely many normals pass through $Q$: both complete circles with $u=\pi /2$ or $3\pi /2$ have radial normals through that point. This is a concrete illustration of degenerate distance-critical sets versus isolated critical points.

On $x^2/a^2+y^2/b^2<1$, the torsion stress function for a simply connected bar is $\varphi =\mu \Omega a^2{b}^2(1-x^2/a^2-y^2/b^2)/(a^2+b^2)$. Its Laplacian is $-2\mu \Omega$ and its boundary value is zero, proving the formula by uniqueness of the Poisson equation with Dirichlet data. Integration gives $M=\pi \mu \Omega a^3{b}^3/(a^2+b^2)$, hence the displayed stiffness $R=M/\Omega$.

The three principal stresses are $0,\pm |\nabla \varphi |$. With ${\tau }_Y$ defined as the shear yield stress, the Tresca yield criterion is $|\nabla \varphi |\le {\tau }_Y$. For $a>b$ its maximum is $2|C|/b$ at $(0,\pm b)$, where $C=\mu \Omega a^2{b}^2/(a^2+b^2)$. Equating that maximum to ${\tau }_Y$ gives the displayed first-yield twist. The ends of the minor axis yield first; geometric distance from the bar axis alone does not determine stress in a noncircular section.

In uniform torsion about the $z$-axis, the axial displacement $w(x,y)$ describes departure of a cross-section from a plane. It is determined up to an additive axial translation by $w_x=\Omega y+{\varphi }_y/\mu$, $w_y=-\Omega x-{\varphi }_x/\mu$, where $\varphi$ is the torsion stress function for a simply connected bar. For a circular section these derivatives vanish. For an ellipse they give $w=\Omega (b^2-a^2)xy/(a^2+b^2)+\text{constant}$, so a noncircular section generally warps. The stress-function Poisson equation is exactly the mixed-derivative compatibility condition for this axial displacement.

The timelike cone is the open set of nonzero tangent vectors with timelike squared norm. It has future and past components once a time orientation is chosen. Homogeneity extends an identity for unit timelike vectors to this whole open set. In particular, if a symmetric quadratic form vanishes on every timelike vector, its polynomial vanishes on an open set and therefore vanishes identically. This observation proves the converse from geodesic first integrals to the Killing equation.

At radius $R$ outside a spherical mass $M$, the Newtonian relative-acceleration tensor has eigenvalues $-g/R,-g/R,2g/R$ with $g=GM/R^2$. It compresses free fall separations tangentially and stretches them radially. The displayed time scale is the inverse square root of the curvature scale controlling this tidal evolution. An ideal circular orbit at that radius has period $2\pi t_{\mathrm{tide}}$, providing a second physical interpretation. For Earth's surface it is about $806$ seconds, while the corresponding ideal low-orbit period is about $84$ minutes. A uniform gravitational acceleration does not determine this relative motion; spatial variation of the field does.

The paraxial image relation for a thin lens. With real-is-positive object and image distances $u$ and $v$ and focal length $f$, it is $1/f=1/u+1/v$. A virtual image has $v<0$ under this convention.

A thin liquid layer whose two reflected beams can interfere. For air on both sides, one surface reflection has a phase reversal; a constructive reflected condition is $2nt\cos r=(j+1/2)\lambda$. Thickness and viewing angle therefore alter the selected colours.

For the leading transport model $H_t+(1+t{)}^{-2}{H}_z=0$, initially dry coating above $z_0$ and matching boundary value $H(z_0,t)=C_0(1+t{)}^{-4/3}$, characteristics preserve $\xi =z+1/(1+t)$. Boundary entry at time $\tau$ gives $F(\xi )=C_0(\xi -z_0{)}^{4/3}$ for $z_0<\xi \le z_0+1$; the remaining initial-data branch is zero. The coated front is $z-z_0=t/(1+t)$. Its final profile is the displayed formula below unit stroke and zero above it. The final scaled film volume is $3C_0/7$, equal to the time-integrated boundary supply. This is a result within the leading transport approximation, not a uniform infinite-time assertion for every finite capillary parameter.

For a symmetric two-dimensional thin drop with uniform evaporative loss $ϵ$, a freely receding contact line and fixed leading outer contact angle $\theta$, quasistatic capillary balance makes the height quadratic. Its volume per transverse length is $2\theta s^2/3$. Since $h_t+(h\overline{u}{)}_x=-ϵ$ and the contact-line flux vanishes, volume loss is $-2ϵs$. On slow time $T=ϵt$, this gives the stated constant recession speed.

Substitute that speed into $h_t$ and integrate flux from the centre, obtaining $h\overline{u}=-ϵx(s^2-x^2)/(4s^2)$. Division by height gives the displayed inward velocity. With a pinned width instead, the same uniform-loss model gives ${\theta }_t=-3ϵ/s$ and outward velocity $\overline{u}=ϵx/(s\theta )$. Thus the contact-line assumption alone reverses solute transport. A pinned perimeter can produce a coffee-ring effect; the receding fixed-angle outer model predicts central transport instead. A strictly no-slip sharp moving contact line is singular, so the finite outer angle is a leading quasistatic approximation requiring microscopic regularization at finite speed.

A thin liquid layer extending ahead of a thicker drop can replace a sharp zero-height contact line in a continuum film model. Keeping the height positive avoids the zero-mobility singularity of a purely no-slip $h^3$ lubrication law at the apparent contact edge. Additional microscopic surface interactions usually determine its thickness and matching to the macroscopic apparent contact angle. It is one possible contact-line regularization, rather than a universal assumption for every substrate.

A thermometer using differential thermal expansion of a liquid and its glass container to move a column through a calibrated capillary.

For an ideal gas with specific internal energy $c_{v}T$ and Fourier's law heat flux $\mathbf{q}=-k\nabla T$, subtraction of the kinetic-energy balance from total energy gives $\rho c_{v}DT/Dt+p\nabla \cdot \mathbf{u}=\nabla \cdot (k\nabla T)$. The thermodynamic entropy satisfies $TDS/Dt=c_{v}DT/Dt+p\nabla \cdot \mathbf{u}/\rho$, soThe divergence describes entropy transport by heat; the final term is local irreversible production. For $T>0$ and $k\ge 0$ it is nonnegative. In a fixed impermeable thermally insulated container, boundary entropy flux vanishes and mass conservation gives $d\int \rho SdV/dt=\int k|\nabla T{|}^2/T^{2}dV$. Insulation conserves total energy while permitting entropy to increase as temperature gradients relax.

In a model of the axioms for the theory of successor, repeatedly taking successor and predecessor partitions the domain into components. Injectivity and the predecessor axiom leave only the zero-generated one-sided chain and two-sided chains; the no-cycle axioms exclude finite components. Choosing one origin in each remaining component gives the displayed structure.

These axioms have quantifier elimination. To eliminate one existential quantification, put its quantifier-free matrix in disjunctive normal form. In a conjunction, an equality between a successor iterate of the bound variable and a parameter term fixes at most one candidate, by injectivity. Existence of the finitely many necessary predecessors excludes only finitely many initial zero iterates; the other equalities and inequalities then translate to conditions on the parameters by cancellation of successor iterates. If there is no such fixing equality, only finitely many candidates are excluded, so some point on the infinite zero chain works. Equalities between two different iterates of the same variable are false by absence of cycles. Thus the existential formula reduces to a quantifier-free formula.

Every closed quantifier-free sentence is decided by the distinct zero iterates, proving completeness. A one-variable formula over the empty set is a Boolean combination of finitely many equalities $x=s^n(0)$, modulo this theory. All points in added integer chains therefore realize the same nonisolated complete type over the empty set. At an uncountable cardinal $\kappa$ a model has exactly $\kappa$ integer components, proving categoricity there. A countable model with infinitely many integer components is an omega-saturated model because finite parameters involve only finitely many components, leaving a fresh one for every new-component type.

The source of the seeds from which cocoa butter is expressed.

The orientation-preserving symmetries of a regular tetrahedron form the alternating group on its four vertices. There are twelve elements: the identity, eight rotational symmetries of order three, and three half-turns.

A contravariant vector has coordinate components obeying the displayed transformation law. A tangent vector to a parametrized curve provides the basic example: the chain rule transforms $d{x}^a/ds$ in precisely this way. Pairing it with a covector produces a coordinate-independent scalar. Conversely, if an array pairs invariantly with every contravariant vector, coefficient comparison forces the covector transformation law for that array.

The linear temporal logic operator $X\alpha$, true at position $i$ of a path in a transition system exactly when $\alpha$ is true at position $i+1$. Thus $XX\alpha$ tests two positions ahead.

In linear temporal logic, $\alpha \mathrm{U}\beta$ holds at position $i$ exactly when some $j\ge i$ satisfies $\beta$ and every position $k$ with $i\le k<j$ satisfies $\alpha$. The witness $j$ must exist. The case $j=i$ imposes no condition on $\alpha$.