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For equal-size subsets $A$ and $B$ of an ordered ground set, $A$ precedes $B$ in lexicographic order when the smallest element of their symmetric difference belongs to $A$.

For finite sequences over a total order, lexicographic order compares the first differing sequence terms; the smaller term determines the smaller sequence. If one sequence is a proper prefix of the other, the shorter precedes its extension. For fixed-length tuples, this is the first-difference rule used by radix sort key encoding.

A bulk-composition ternary diagram represents the mass fractions of three planetary components at a point whose barycentric coordinates sum to one.

Exoplanet habitability concerns whether planetary conditions could sustain liquid water and life, while recognizing that observability and habitability are distinct questions.

An exoplanet atmosphere controls the planet's spectrum, heat redistribution, chemistry, clouds, and atmospheric escape.

For each positive integer $k$, only finitely many $n$ satisfy $\phi (n)=k$. Indeed, every prime $p\mid n$ has $p-1\mid k$, and the exponent of $p$ in $n$ is bounded because $p^{a-1}\mid k$.

For coprime positive integers m and n, the Chinese remainder theorem gives phi(mn)=phi(m)phi(n).

If a is coprime to n, then a raised to the power phi(n) is congruent to one modulo n.

Multiplication by a unit $a$ permutes the reduced residue classes modulo $n$. Multiplying all classes before and after this permutation and cancelling their unit product proves $a^{\varphi (n)}\equiv 1(\mathrm{mod}n)$.

Forindependent variations and integration by parts give

When the values of $y$ are free at both fixed endpoint locations, integration by parts leaves the boundary term $[F_{y^{\prime }}\eta {]}_a^b$. Since the endpoint variations are arbitrary, stationarity requires

A catenary is a curve of the form $y=a\cosh ((x-x_0)/a)+y_0$. It is the equilibrium shape of a uniform flexible chain in a uniform gravitational field.

When a variational integrand $f(z,z^{\prime })$ has no explicit dependence on the independent variable, every extremal satisfies

The integral momentum equation balances momentum flux, pressure, body force, and forces on a control volume.

For steady inviscid flow, pressure, kinetic energy density, and conservative potential are constant along a streamline.

For $\mathbf{u}=\nabla \varphi +\beta \nabla \alpha$, the densitygives incompressibility and material conservation of $\alpha$ and $\beta$. Differentiating the Clebsch representation then yields the Euler momentum equation with

For time-dependent irrotational flow with $u=\nabla \varphi$, constant mass density, and no body force, the Euler equations for an inviscid fluid integrate toA time-dependent change of gauge in $\varphi$ may set $C(t)=0$.

This is the unsteady potential-flow version of Bernoulli's principle, with a time-dependent potential gauge.

A lattice polygon has all vertices in the integer lattice; triangulation shows that its area is a half-integer.

A spherical cap is the part of a sphere cut off by a plane.

A regular polygon has equal side lengths and equal interior angles. Its vertices on a circle are scaled and rotated roots of unity.

A right angle is an angle of $\pi /2$ radians; two nonzero vectors meet at a right angle exactly when their inner product is zero.

The perpendicular bisector of a line segment is the line through its midpoint at a right angle to the segment.