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The circle is the subspace $S^1=\{z\in \mathbb{C}:|z|=1\}$, equivalently the quotient of a closed interval obtained by identifying its endpoints.

The unitary group $U(n)$ consists of complex matrices preserving the standard Hermitian inner product.

For loops based at the identity of a topological group, pointwise multiplication and concatenation obey the interchange law and have the same identity. The Eckmann-Hilton argument therefore makes the fundamental group abelian.

For a two-component gapped BdG Hamiltonian constrained to a plane, the winding number counts how many times its coefficient vector encircles the origin as crystal momentum traverses the Brillouin zone.

When pressure and velocity are in quadrature, their instantaneous energy flux oscillates but its period average vanishes.

For $N$ identical sites of spacing $a$ with periodic boundary conditions, translation eigenstates have coefficientsThus $k=2\pi j/(Na)$ modulo the reciprocal-lattice period $2\pi /a$.

The Hills mass is the black-hole mass above which a given star crosses the capture boundary before being tidally disrupted. For a Schwarzschild capture radius $2G{M}_{\mathrm{BH}}/c^2$, it is $c^3{R}_{\ast }^{3/2}/(2G^{3/2}{M}_{\ast }^{1/2})$ with the corresponding tidal-radius convention.

For a star of mass $M_{\ast }$ and radius $R_{\ast }$ near a black hole of mass $M_{\mathrm{BH}}$, the Newtonian tidal disruption radius is $r_t\simeq R_{\ast }(2M_{\mathrm{BH}}/M_{\ast }{)}^{1/3}$ under a peak-to-center tidal-acceleration convention.

The surface density of a disk is the mass density integrated through its vertical column: $\Sigma (r)={\int }_{-\infty }^{\infty }\rho (r,z)dz$. Its dimensions are mass per area. An axisymmetric annulus of width $dr$ therefore contains $dM=2\pi r\Sigma (r)dr$. This reduces a three-dimensional mass inventory to a radial one, which is useful in viscous evolution of an accretion disk. Distinct vertical profiles can have the same column mass, so $\Sigma$ alone does not determine the vertical structure of an astrophysical disk.

The ideal Diesel cycle consists of reversible adiabatic compression, constant-pressure heat addition, reversible adiabatic expansion, and constant-volume heat rejection.

The ideal Otto cycle consists of reversible adiabatic compression, constant-volume heat addition, reversible adiabatic expansion and constant-volume heat rejection. For a fixed amount of ideal gas with constant heat capacity ratio $\gamma =C_p/C_V>1$ and compression ratio $r=V_1/V_2>1$, its thermal efficiency isThe reversible adiabats are isentropic processes; merely preventing heat exchange would not make an irreversible compression obey the same relations.

To derive the formula, number the four states in cycle order. The adiabats giveWith constant whole-sample heat capacity at constant volume $C_V$, the absorbed and rejected heat magnitudes areSubstitution into $\eta =1-Q_{\mathrm{out}}/Q_{\mathrm{in}}$ cancels the heat-addition temperature difference and yields the stated result.

Within this model, larger $r$ increases thermal efficiency. Real engines also depend on combustion, heat transfer, friction and the temperature dependence of heat capacities; the formula does not predict those losses.

The thermal efficiency of a heat engine is the net work output divided by heat input. For a cycle absorbing $Q_{\mathrm{in}}$ and rejecting the positive amount $Q_{\mathrm{out}}$,

If a relativistic species decouples when the coupled plasma has $g_{\ast s,\mathrm{dec}}$ entropy degrees of freedom, then after the plasma reaches $g_{\ast s}$ degrees of freedom,The ratio follows from separate redshifting of the decoupled species and cosmological entropy conservation in the coupled plasma.

For $u(y)=Uy/h$ between reflecting walls at $y=\pm h$, the flow-induced diffusivity is

The velocity vector of a differentiable parametrized curve $r(t)$ is its tangent vector $r^{\prime }(t)$ at the base point $r(t)$. Its norm, when a metric is specified, is the speed. For $r(t)=(\cos t,\sin t)$ on the unit circle, $r^{\prime }(t)=(-\sin t,\cos t)$ has unit speed and is perpendicular to the radius. Replacing $t$ by $2t$ doubles speed without changing the image curve. Thus a geometric route and its timed traversal carry different information.

For a regular curve $\mathbf{r}(t)$ with nonzero velocity vector, its unit tangent vector isThe Euclidean norm in the denominator removes the traversal speed, leaving a unit vector that records direction. A positive change of traversal speed leaves this direction unchanged; reversing the direction of travel reverses $T$. In arc length coordinates, $T=d\mathbf{r}/ds$.

For the parabola $\mathbf{r}(t)=(t,t^2)$, the velocity vector is $(1,2t)$ and the speed is $\sqrt{1+4t^2}$. At $t=1$, $T=(1,2{)}^T/\sqrt{5}$. The tangent vector $(1,2{)}^T$ and the unit tangent vector point in the same direction, but only the latter has Euclidean norm one. The principal normal vector is a further choice based on how this direction changes, where the curvature is nonzero. ^[ref]

At a fixed point $p$ of an $n$-dimensional smooth manifold $M$ without boundary, the tangent space $T_{p}M$ is the real vector space of all tangent vectors at $p$. In manifold charts $x=(x^1,\dots ,x^n)$ and $y=(y^1,\dots ,y^n)$, a tangent vector has componentsThe coordinate basis transforms inversely, so the tangent vector is unchanged. This invertible Jacobian matrix also preserves vector addition and scalar multiplication, giving a vector space of dimension $n$ independently of coordinates. ^[ref]

For example, at $(x,y)=(1,0)$ in ${\mathbb{R}}^2$, use the alternative manifold chart $u=x$, $w=y+x^2$. The Jacobian matrix sends $(v_x,v_y)=(2,3)$ to $(v_u,v_w)=(2,7)$. Indeed, the smooth curve $(x(t),y(t))=(1+2t,3t)$ gives $w(t)=1+7t+4t^2$. Conversely, ${\partial }_u{|}_p={\partial }_x{|}_p-2{\partial }_y{|}_p$ and ${\partial }_w{|}_p={\partial }_y{|}_p$, so $2{\partial }_u{|}_p+7{\partial }_w{|}_p=2{\partial }_x{|}_p+3{\partial }_y{|}_p$. The numbers change; their action on every smooth function does not.

For an embedded submanifold $M\subseteq {\mathbb{R}}^N$, the tangent space can be identified with the vector subspaceA proof outline explains why every coordinate vector is possible: in a manifold chart $x$, lift the short straight path $x(p)+ta$ using $x^{-1}$. Its tangent vector has components $a$. For a regular local parametrization $X:U\subseteq {\mathbb{R}}^n\to M$, the same chain rule gives $T_{p}M=\mathrm{im}D{X}_q$, where $X(q)=p$. This is the tangent space from a local parametrization.

For the sphere $S_R^2=\{p\in {\mathbb{R}}^3:p\cdot p=R^2\}$, differentiating that equation gives $p\cdot v=0$. Conversely, for any nonzero $v$ with $p\cdot v=0$, the smooth curvelies on the sphere and has ${\gamma }_v^{\prime }(0)=v$; use a constant smooth curve for $v=0$. Thus $T_p{S}_R^2=\{v:p\cdot v=0\}$, not merely a subset of that plane. At $p=(0,0,R)$ this vector subspace is $v_z=0$, while the affine tangent plane drawn through $p$ is $z=R$.

This also explains a mechanical constraint without requiring a force calculation. In the ideal mathematical model of a particle confined to a stationary sphere, its allowed velocities at $p$ are exactly $T_p{S}_R^2$: the radial component must vanish, but either independent tangential direction is available. For a general stationary holonomic constraint $F(q)=0$ with independent equations, the regular level set theorem gives $T_{q}M=\ker D{F}_q$. These are instantaneous kinematic possibilities; they do not determine the actual subsequent motion. On a general smooth manifold, tangent spaces at different base points cannot be identified canonically; an affine connection supplies parallel transport when such comparisons are needed.

At the identity, the symplectic Lie algebra consists of $X$ satisfying $X^{T}J+JX=0$; tangent spaces elsewhere are its left translates.

The map $F:M_{2n}(\mathbb{R})\to {\mathrm{Skew}}_{2n}(\mathbb{R})$ given by $F(A)=A^{T}JA$ has $Sp(2n,\mathbb{R})=F^{-1}(J)$. At a symplectic $A$,is surjective, so the regular level set theorem makes the symplectic group an embedded submanifold.

The one-dimensional cluster state is the common positive eigenstate of commuting stabilizers $Z_{j-1}{X}_j{Z}_{j+1}$ and realizes a nontrivial ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ SPT phase.