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A planetary dynamo maintains an intrinsic magnetic field by motion of an electrically conducting fluid. The resistive induction equation balances stretching and transport against magnetic diffusion. A sufficiently large magnetic Reynolds number is necessary but does not ensure that the flow regenerates every required field component. Thermal or compositional convection supplies mechanical power; rotation organizes the flow through the Coriolis force, and the Lorentz force density eventually modifies the flow and limits growth. A fluid core can fail to operate a dynamo if it is stably stratified or inadequately driven.

The innermost planet has a large metallic core and a weak but global intrinsic magnetic field. Its observed field supports an active planetary dynamo in a fluid conducting part of that core. Slow rotation does not by itself exclude dynamo action.

Two viscous layers occupy $-a<y<0$ and $0<y<a$, with viscosities ${\mu }_{-}$ and ${\mu }_{+}$. A uniform driving pressure gradient $G=-p_x>0$, a stationary flat interface and no-slip outer walls give the displayed plane Poiseuille flow, where $U_0=G{a}^2/({\mu }_{-}+{\mu }_{+})$ and $C=Ga({\mu }_{-}-{\mu }_{+})/[2({\mu }_{-}+{\mu }_{+})]$. These constants follow from continuity of velocity and shear stress at the interface and zero wall velocities. The upper-to-lower volume-flux ratio is $(1+7r)/[r(7+r)]$, with $r={\mu }_{+}/{\mu }_{-}$. As $r\to \infty$, the interface becomes effectively stationary, the upper flux vanishes, and the lower layer approaches flow between two stationary plates a distance $a$ apart. The upper wall stress remains finite despite its vanishing velocity gradient.

At large Reynolds number $\mathcal{R}$, a long indentation of relative height $ϵ$ has nonlinear near-wall interaction when the streamwise scale ratio satisfies $\lambda /\mathcal{R}={ϵ}^3$. The regime ${\mathcal{R}}^{1/7}\ll \lambda \ll \mathcal{R}$ gives a thin viscous layer and an effectively common pressure across the core. The core displacement is $u_1=A(x)U_0^{\prime }(y)$ and $v_1=-A^{\prime }(x)U_0(y)$. A lower wall displaced to $ϵF(x)$ produces layer displacement $F+A$, while the upper wall produces $-A$. Uniqueness for their common pressure forcing and upstream data makes the two layer displacements equal, so $A=-F/2$.

A steady planar viscous jet whose conserved momentum flux and convection-diffusion balance give the displayed scaling. Choosing $U{\delta }^2=\nu x$ and $U$ as centre-line speed gives streamfunction profile $f^{\prime \prime \prime }+(f{f}^{\prime \prime }+(f^{\prime }{)}^2)/3=0$, with $f(0)=f^{\prime \prime }(0)=0$, $f^{\prime }(0)=1$ and $f^{\prime }(\pm \infty )=0$. Integration yields $f=\sqrt{6}\tanh (\eta /\sqrt{6})$ and velocity profile $f^{\prime }={\mathrm{sech}}^2(\eta /\sqrt{6})$. Its momentum integral is $4\sqrt{6}/3$, fixing the dimensional prefactors. Transverse entrainment does not require $f(\infty )=0$.

For $d{s}^2=-2dudv+d{x}^2+d{y}^2-2H(u,x,y)d{u}^2$, the nonzero connections are ${\Gamma }_{uu}^v=H_u$, ${\Gamma }_{ux}^v=H_x$, ${\Gamma }_{uy}^v=H_y$, ${\Gamma }_{uu}^x=H_x$, and ${\Gamma }_{uu}^y=H_y$, with symmetric lower indices. Their traces and quadratic Ricci products vanish, leaving only the displayed Ricci component. Hence the vacuum equations require a harmonic transverse profile. The sign is tied to the metric's $-2Hd{u}^2$ convention.

Across an interface with unit normal $\mathbf{n}$ from medium one to medium two, Maxwell equations imply $\mathbf{n}\times ({\mathbf{E}}_2-{\mathbf{E}}_1)=0$, $\mathbf{n}\cdot ({\mathbf{B}}_2-{\mathbf{B}}_1)=0$, $\mathbf{n}\cdot ({\mathbf{D}}_2-{\mathbf{D}}_1)={\sigma }_f$, and $\mathbf{n}\times ({\mathbf{H}}_2-{\mathbf{H}}_1)={\mathbf{K}}_f$. Here ${\sigma }_f$ and ${\mathbf{K}}_f$ are the free surface electric charge and electric current densities.

A figure-eight plane curve with two congruent lobes and a crossing at the origin. With scale $a>0$, its polar equation is $r^2=a^2\cos 2\theta$. One quarter has coordinates $x=at\sqrt{(1+t^2)/2}$, $y=at\sqrt{(1-t^2)/2}$, $0\le t\le 1$. Differentiation gives arc element $adt/\sqrt{1-t^4}$. The total arc length is therefore $4a{\int }_0^{1}dt/\sqrt{1-t^4}$.

Reduced Planck units choose $\hslash =c=8\pi G=1$. With the reduced Planck mass defined by $M_{\mathrm{Pl}}=(8\pi G{)}^{-1/2}$ after setting $\hslash =c=1$, this convention gives $M_{\mathrm{Pl}}=1$ and the Einstein-Hilbert action coefficient $M_{\mathrm{Pl}}^2/2=1/2$.

For a spatially flat Friedmann-Lemaître-Robertson-Walker metric, the Friedmann equation is then $3H^2=\rho$, where $\rho$ is the total energy density, including any vacuum contribution. With spatial-curvature parameter $k$, it is instead $3(H^2+k/a^2)=\rho$. Setting the unreduced Planck mass to one would set $G=1$, leaving the factor $8\pi$ in the gravitational equation. Keeping these conventions distinct prevents spurious factors in gravitational and inflationary calculations.

For $h\in C_c(G)$, apply Fourier inversion on a locally compact abelian group at the identity to $h\ast h^{\ast }$, whose transform is $|\hat{h}{|}^2$. This proves the displayed norm identity on $C_c(G)$. Density and completeness give a unique linear isometric extension to all $L^2(G)$. On $L^1\cap L^2$ it agrees almost everywhere with the integral Fourier transform on a locally compact abelian group.

With the Fourier convention $\hat{f}(t)=\int f(x)e^{-itx}dx$, translation by $k$ multiplies the transform by $e^{ikt}$. The Plancherel theorem gives the displayed correlation identity for every square-integrable function, by extension from integrable square-integrable functions. Thus distinct integer translates are orthogonal exactly when the nonzero integer Fourier integrals of $|\hat{f}{|}^2$ vanish. The products in both integrals are integrable by the Cauchy-Schwarz inequality.

Every finite planar graph admits a proper vertex coloring with at most five colors. Euler formula for a connected planar graph supplies a vertex of degree at most five. Inductively color its deletion. The only obstruction is five differently colored neighbours in cyclic order. If opposite neighbours 1 and 3 are in different two-color components, interchange one component. If they are connected, planarity prevents opposite neighbours 2 and 4 from being connected in their two-color subgraph, and that interchange frees a color instead. This is a Kempe chain proof.

Every finite planar graph has chromatic number at most four. This stronger theorem is distinct from the elementary five colour theorem; the latter's Kempe chain proof does not prove this assertion.

The honeycomb lattice is the periodic degree-three planar graph consisting of the vertices and edges of a regular hexagonal tiling. It has two vertex classes; every edge joins opposite classes. Its planar dual graph is the triangular lattice.

A plane graph is a planar graph with a specified graph drawing whose edges meet only at common endpoints, also called an embedding in the Euclidean plane. Its faces of a plane graph and facial boundary walks refer to that embedding. This geometric use of embedding differs from a graph embedding, which maps one abstract graph into another.

A finite simple graph is a planar graph if and only if it contains no graph subdivision of the complete graph $K_5$ or the complete bipartite graph $K_{3,3}$ as a subgraph; equivalently, neither is a topological minor. The easy direction follows by tracing each replacement graph path as one drawn edge. The converse asserts that every failure of planarity contains one of these two patterns.^[ref]

Graph Theory by Reinhard Diestel (Planar Graphs preview, §4.4, Theorem 4.4.6, printed p. 111) dates Kuratowski's theorem to 1930 and the equivalent graph minor characterization to 1937. These dates are that textbook's historical attribution.

In the displayed pitchfork bifurcation normal form, zero is stable for $\mu <0$ and unstable for $\mu >0$. At $\mu =0$, a symmetric stable pair $x=\pm \sqrt{\mu }$ appears on the side where zero is unstable. Reversing the parameter orientation changes the side, while retaining the supercritical relationship between the new stable branches and the destabilized central branch.

The polar form of the cubic confinement model has an origin pitchfork on ${\mu }^2+{\sigma }^2=1$ when $\mu \ne 0$. Its small squared amplitude satisfies $S\sim 2\mu \delta \mu /(\mu +\sigma )$ along fixed $\sigma$. Thus the leading cubic coefficient vanishes when $\mu +\sigma =0$, at $(\sigma ,\mu )=(\pm 1/\sqrt{2},\mp 1/\sqrt{2})$. At either point the exact radius equation gives $\delta \mu =-S^2/(4{\mu }_0)+o(S^2)$, so amplitude scales as $|\delta \mu {|}^{1/4}$. The lower point has a stabilizing quintic coefficient $1/(4{\mu }_0)<0$; folds of nonzero equilibrium pairs meet the pitchfork curve there. The upper point has the opposite quintic sign and an already unstable transverse eigenvalue.

For $k>1$, the rational equilibrium point curve in the display has expansion $r=1+q+[1-q(k-1)]u+qk(k-1)u^2+\cdots$. Its cubic pitchfork bifurcation coefficient vanishes at $q=1/(k-1)$, leaving $r-(1+q)=k{u}^2+\cdots$. For larger $q$, the nonzero branch folds at $u=[\sqrt{q(k-1)}-1]/k$ and $r=[k-1+q+2\sqrt{q(k-1)}]/k$. If the other eigenvalues remain stable, the two folds bound a bistable wedge between stable zero and stable nonzero states, separated by an unstable branch. If another eigenvalue is critical, a higher-dimensional centre manifold is required instead.

The Hagen-Poiseuille equation describes steady, fully developed, laminar flow of an incompressible Newtonian fluid through a straight rigid circular pipe. With pipe radius $R$, axial coordinate $z$ and constant viscosity $\mu$, force balance reduces to $\mu r^{-1}\frac{d}{dr}(r\frac{dw}{dr})=dp/dz$.

Steady pressure-driven flow through a circular pipe has the parabolic profile

Regularity at the centre and the no-slip boundary condition at $r=R$ fix the displayed profile. Integrating over the cross-section gives $Q=-\pi R^4(dp/dz)/(8\mu )=\pi R^4\Delta p/(8\mu L)$ for length $L$ and positive pressure drop $\Delta p$. The fourth power makes volume flux strongly sensitive to radius; the assumptions exclude arbitrary turbulent or developing pipe flow. ^[ref]