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For a small body with orbital elements $a,e,I$ relative to a planet on a circular orbit of semi-major axis $a_p$, the Tisserand parameter is approximatelyIt is derived from the Jacobi constant and is approximately conserved across separated weak encounters in the circular restricted three-body problem.

A planetesimal is a solid body in a young or evolved planetary system whose collisions and gravitational interactions help build or erode larger bodies.

For $D_i={\epsilon }_{ij}{E}_j$, $B=\mu H$, and a plane wave with electric polarization $e$, the source-free Maxwell equations implyIf $e$ is an eigenvector of $\epsilon$, then $e\perp k$, the dispersion relation is $|k{|}^2={\omega }^2\mu {\epsilon }_e$, and the Poynting vector is parallel to $k$.

Without free surface charge or current, the tangential components of $E$ and $H$ and the normal components of $D$ and $B$ are continuous across an interface.

At the surface of a perfect conductor, the tangential electric field and normal magnetic field vanish. An incident plane electromagnetic wave is therefore accompanied by a reflected wave whose tangential electric field cancels the incident field at the surface.

A semicubical parabola is a cusped cubic curve affinely equivalent to $y^2=x^3$.

For a smooth plane curve of degree $d\ge 4$, adjunction givesIts canonical map is induced by the $(d-3)$rd Veronese map and is therefore an embedding.

For a finite connected graph with at least one vertex, embedded as a plane graph, let $n$ count vertices, $m$ count edges, and $f$ count faces of a plane graph, including the outer face. ThenThe hypotheses refer to an actual embedding; regions in a crossing-filled graph drawing do not supply its $f$.^[ref]

In the plane graph $K_4$ pictured above, three bounded faces of a plane graph and the outer face give $4-6+4=2$. A square cycle graph gives $4-4+2=2$, while the graph path $A-B-C$ gives $3-2+1=2$. Changing a crossing into a new vertex changes the graph: the square-and-diagonals drawing then has five vertices, eight edges, and five faces of a plane graph, giving $5-8+5=2$.

A proof outline deletes edges on graph cycles. Each deletion preserves the connected graph and merges two faces of a plane graph, leaving $n-m+f$ unchanged. Eventually a tree remains, with $m=n-1$ and $f=1$. Adding the point at infinity identifies the plane picture with a sphere; the value two agrees with its Euler characteristic and the cell-count formula for Euler characteristic.

A planar dual graph has one vertex in each face of a fixed planar embedding and one dual edge crossing each primal edge.

This is the planar-embedding case of a dual graph.

For a place $v$ of a number field $K$, the local degree is $d_v=[K_v:{\mathbb{Q}}_v]$. Thus $d_v=1$ at a real place, $d_v=2$ at a complex place, and $d_v=e_v{f}_v$ at a finite place over $p$. For each rational place, the local degrees above it sum to $[K:\mathbb{Q}]$.

If a prime ideal $\mathfrak{p}$ of $K$ lies over the prime number $p$ with ramification index of a prime ideal $e_{\mathfrak{p}}$, its normalized absolute value isIt extends the usual p-adic absolute value on $\mathbb{Q}$. Its local degree is $[K_{\mathfrak{p}}:{\mathbb{Q}}_p]=e_{\mathfrak{p}}{f}_{\mathfrak{p}}$, where $f_{\mathfrak{p}}$ is the residue-field degree.

A real embedding $K\hookrightarrow \mathbb{R}$ gives a real Archimedean place of local degree $1$. A conjugate pair of complex embeddings $K\hookrightarrow \mathbb{C}$ gives a complex Archimedean place of local degree $2$.

Reflection symmetry $x\mapsto -x$ forces the reduced vector field to be odd in $x$, naturally producing paired nonzero equilibrium branches.

The normal form $\dot{x}=\mu x+x^3$ has a stable trivial equilibrium for $\mu <0$ and two unstable nonzero equilibria for $\mu <0$. They collide with the trivial branch at $\mu =0$, after which that branch is unstable.

The photon quadrupole is the leading anisotropy that sources linear polarization through Thomson scattering.

The photon dipole represents the bulk velocity of the radiation. In tight coupling to baryons, $v_b=-3{\Theta }_1$ under the convention used here.

The photon monopole is the direction-averaged fractional temperature perturbation. In Newtonian gauge its gravitationally observable effective temperature is the Sachs-Wolfe combination ${\Theta }_0+\Psi$.

The comoving sound horizon is the distance traveled by an acoustic wave,

The tight-coupling approximation expands the photon-baryon Boltzmann equations in powers of $k/\Gamma$ and $\mathcal{H}/\Gamma$, where $\Gamma$ is the Thomson scattering rate.

A symplectic form is a closed differential form of degree two whose value on every tangent space is a nondegenerate alternating bilinear form. In canonical position-momentum coordinates it is ${\sum }_{i}d{q}_i\wedge d{p}_i$.