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A differential one-form is closed when its exterior derivative vanishes. In coordinates, $\eta ={\eta }_{i}d{x}^i$ is closed exactly when ${\partial }_i{\eta }_j={\partial }_j{\eta }_i$. The Poincaré lemma makes it locally an exact differential form, but global exactness can fail through nonzero integrals around closed curves.

For a closed convex cone $C$ in a finite-dimensional real inner product space and $z\notin C$, there exists $y$ with $⟨y,z⟩<0$ and $⟨y,c⟩\ge 0$ for all $c\in C$. Here is a direct proof. A nearest point $c_0\in C$ exists by compactness after restricting to a sufficiently large ball. Differentiating squared distance along the segment towards $c\in C$ gives $⟨c_0-z,c-c_0⟩\ge 0$. Using $c=0$ and $c=2c_0$ shows $⟨c_0-z,c_0⟩=0$. Set $y=c_0-z\ne 0$: then $⟨y,c⟩\ge 0$ and $⟨y,z⟩=-\Vert y{\Vert }^2<0$. This is a conic form of the Hahn-Banach separation theorem.

For the number $X$ of $k$-cliques in a binomial random graph $G(n,p)$, with $0<p<1$ and $\mu =\mathbb{E}X$,Two $k$-sets overlapping in $j$ vertices share $\left(\genfrac{}{}{0ex}{}{j}{2}\right)$ edges. Overlaps of zero or one vertex give independent indicators; $j=k$ includes the diagonal terms. The identity is useful for the second moment method.

The clique number $\omega (G)$ is the maximum number of vertices in a clique of a finite graph:We use $\omega (G)=0$ for the graph with no vertices. For a nonempty complete graph, $\omega (K_n)=n$; for the five-vertex example, $\omega (G)=3$.

If the graph contains $K_k$ and no $K_{k+1}$ as subgraphs, its clique number is exactly $k$, because every larger clique contains a clique with $k+1$ vertices. Also $\chi (G)\ge \omega (G)$: a proper vertex coloring must assign different colors to all members of any clique. This lower bound need not be an equality; the cycle graph $C_5$ has clique number $2$ and chromatic number $3$.

An edge-disjoint clique packing is a collection of $r$-cliques no two of which share an edge; they may share vertices. Its maximum cardinality $Z_r(G)$ changes by at most one when a single edge is toggled. Removing that edge destroys at most one packed clique, which makes this variable suitable for an edge-exposure martingale.

A symmetrized form of the Clebsch-Gordan coefficients used for coupling three angular momenta. The relation is $⟨j_1{m}_1,j_2{m}_2|j_3{m}_3⟩=(-1{)}^{j_1-j_2+m_3}\sqrt{2j_3+1}\left(\begin{array}{ccc}{j}_1& j_2& j_3\\ m_1& m_2& -m_3\end{array}\right)$. The symbol vanishes unless the magnetic indices sum to zero and the angular momenta obey the triangle and integrality conditions. Products of two such symbols express the Gaunt integral; the zero-magnetic-index symbol enforces even total multipole for integer spherical harmonics.

For $u\in H_0^2(U)$, ${\sum }_{i,j}\Vert {\partial }_{ij}u{\Vert }_2^2=\Vert \Delta u{\Vert }_2^2$. Two integrations by parts prove this for compactly supported test functions, and $H^2$ density extends it to the clamped second-order Sobolev space. Together with the Poincare-Wirtinger inequality applied to the mean-zero first derivatives and the zero-boundary Poincaré inequality, it controls the full $H^2$ norm by $\Vert \Delta u{\Vert }_2$. The boundary conditions are essential to this exact identity on bounded domains.

For ${\zeta }_{\perp }{h}_t=-B{h}_{xxxx}$, separated normal modes decay at rates $B{\alpha }_n^4/({\zeta }_{\perp }{L}^4)$. Clamping at zero and vanishing moment and shear at $L$ give $\cos {\alpha }_n\cosh {\alpha }_n=-1$. A mode isThe fourth-order operator is self-adjoint: two integration by parts operations give $\int f{g}^{\prime \prime \prime \prime }=\int f^{\prime \prime }{g}^{\prime \prime }$ under these boundary conditions. Its zero eigenspace is trivial and its eigenfunctions form an orthogonal complete basis. Thus a nonzero projection on the first mode gives long-time decay proportional to ${\varphi }_1(x/L)\exp [-B{\alpha }_1^{4}t/({\zeta }_{\perp }{L}^4)]$, where ${\alpha }_1=1.8751040687\dots$. If this projection vanishes, the lowest excited mode replaces it.

The elastic energy $U=A\int (W^{\prime \prime }{)}^{2}dx/2$ minimized without forcing gives $W^{\prime \prime \prime \prime }=0$. With fixed $\int W^{2}dx$, stationarity of $U-A{k}^4\int W^{2}dx/2$ gives $W^{\prime \prime \prime \prime }=k^{4}W$: a bending eigenmode. A general admissible shape expands in these normal modes. The natural boundary conditions for a free endpoint are $W^{\prime \prime }(L)=W^{\prime \prime \prime }(L)=0$, while the clamp imposes $W(0)=W^{\prime }(0)=0$.

For the positive roots $q_n$ of $\cos q\cosh q=-1$, ${\sum }_n{q}_n^{-4}=1/12$. The tip-force compliance of a cantilever is $L^3/(3A)$ directly, whereas expansion in normalized clamped--free bending modes and $W_n(L{)}^2=4L^{-1}\int W_n^{2}dx$ gives $4L^3{\sum }_n{q}_n^{-4}/A$. Equating the two responses proves the sum and evaluates the full thermal endpoint variance.

The leading uniform stellar radiation pressure is a differential acceleration relative to a planet, of scale $\beta G{M}_{\star }/a_p^2$. Relative to planetocentric gravity at dust radius $a$, its magnitude isThus lying inside a Hill sphere alone does not guarantee small radiative perturbations. Approximately circular tightly bound dust requires this ratio sufficiently small, as well as weak tides; the uniform radiative force can otherwise excite orbital eccentricity or unbind grains.

A circumplanetary orbit whose orbital period equals the central planet's spin period. Its semi-major axis follows from Kepler's third law. Matching periods alone does not fix the satellite above one ground location; that additionally requires a planet-stationary orbit.

A circular cylinder of radius $\rho$ has induced metric locally equal to $d{u}^2+d{z}^2$ with $u=\rho \phi$. Its Riemann curvature tensor therefore vanishes, despite its nonzero extrinsic curvature. With outward unit normal and convention $K_{ij}=e_i^a{e}_j^b{\nabla }_a{n}_b$, its trace is $1/\rho$ and its principal curvatures are $1/\rho$ and zero. Its averaged mean curvature is $1/(2\rho )$. The angular identification changes global topology, not local flatness.

Place equal numbers of radii $r_1,r_2$ on alternating sites of a square nearest-neighbour lattice of spacing $d$. Nonoverlap requires $d\ge \max (r_1+r_2,\sqrt{2}{r}_1,\sqrt{2}{r}_2)$. A two-circle cell has area $2d^2$, giving the displayed packing fraction. Its maximum within this geometry occurs at large-to-small radius ratio $1+\sqrt{2}$ and is $\pi (1-1/\sqrt{2})$; smaller interstitial circles leave unused space, while larger ones force the large-circle lattice apart.

The radius of a circle or sphere is the Euclidean distance from its centre to its boundary. The diameter is twice the radius.

An equation $|az+b\overline{z}|=ux+v$, with $z=x+iy$ and real coefficients, becomes $(a+b{)}^2{x}^2+(a-b{)}^2{y}^2=(ux+v{)}^2$ after squaring. If $(a+b{)}^2-u^2=(a-b{)}^2>0$, completing the square gives a circle. The original locus is only the part where $ux+v\ge 0$. Checking this sign on the candidate circle prevents extraneous points introduced by squaring. This method combines the complex modulus with elementary conic geometry.

A vertex $v$ is colour-critical if deleting it lowers the chromatic number by one. If $\chi (F)=r+1$ and $\chi (F-v)=r$, an embedding of $F-v$ into a complete $r$-partite graph inside one neighbourhood extends to an embedding of $F$. This supplies maximum-degree obstructions in extremal graph theory.

For a nondegenerate metric tensor and its Levi-Civita connection, contracting the Christoffel symbol formula gives ${\Gamma }^{\rho }{}_{\rho \mu }=\frac{1}{2}{g}^{ab}{\partial }_{\mu }{g}_{ab}$. The Jacobi determinant derivative formula identifies this with ${\partial }_{\mu }\log \sqrt{|\det g|}$. The volume density, rather than the determinant without an absolute value, works in every fixed metric signature.

A sum of the squares of $n$ independent random variables with the standard normal distribution has the chi-squared distribution with $n$ degrees of freedom. A scaled Gaussian integral and factorization by independence give its moment-generating function $(1-2t{)}^{-n/2}$ for $t<1/2$. For positive $n$, this expectation diverges when $t\ge 1/2$.

For $V\sim {\chi }_n^2$ and $x>0$, the upper tail is at most $e^{-x}$ above $n+2\sqrt{nx}+2x$, and the lower tail is at most $e^{-x}$ below $n-2\sqrt{nx}$. Indeed $\log \mathbb{E}{e}^{s(V-n)}=-ns-(n/2)\log (1-2s)\le n{s}^2/(1-2s)$ for $0<s<1/2$. The Chernoff bound with $s=\sqrt{x/n}/(1+2\sqrt{x/n})$ proves the upper estimate. For the lower estimate, $\log \mathbb{E}{e}^{-s(V-n)}\le n{s}^2$ for $s>0$, and optimization gives the result. The lower threshold can be negative; the chi-squared Chernoff lower-tail bound provides an always-positive alternative.