If $\sum a_n$ converges and $b_n$ is bounded with $\sum |b_{n+1}-b_n|<\infty$, then $\sum a_n{b}_n$ converges. For $T_j={\sum }_{k=m}^j{a}_k$, summation by parts givesThe first factor tends uniformly to zero by the Cauchy sequence criterion, and the second has a uniform bound. A bounded monotone sequence has finite total variation, so monotone multipliers are included. Eventual monotonicity also suffices, since finitely many initial terms do not affect convergence. Absolute convergence of $\sum a_n$ is unnecessary.

For a positive lapse function $\alpha$, shift vector ${\beta }^i$ and spatial induced metric ${\gamma }_{ij}$, the block metric has $g_{00}=-{\alpha }^2+{\gamma }_{ij}{\beta }^i{\beta }^j$ and $g_{0i}={\gamma }_{ij}{\beta }^j$. Its block determinant is $g=-{\alpha }^2\det \gamma$. Therefore the metric volume tensor contracted with the future unit normal restricts to the spatial volume tensor: $n^{\mu }{ϵ}_{\mu ijk}=\sqrt{\det \gamma }[ijk]$.

In fixed spatial coordinates set $a_{\mathrm{loc}}=(\det h{)}^{1/6}$ and ${\gamma }_{ij}=a_{\mathrm{loc}}^{-2}{h}_{ij}$, so $\det \gamma =1$. This volume-shape decomposition is useful for separating isotropic expansion and anisotropic shape. The determinant-defined scale depends on the coordinate volume convention.

For $P=W^{\prime }$ and $S=|P{|}^2-P^{\prime }{\theta }_1{\theta }_2-\overline{P^{\prime }}{\bar{\theta }}_1{\bar{\theta }}_2$, four odd graded derivations preserve $S$: ${\theta }_1{\partial }_z-\bar{P}{\partial }_{{\theta }_2}$, ${\theta }_2{\partial }_z+\bar{P}{\partial }_{{\theta }_1}$ and their barred counterparts. Left Grassmann derivatives fix the signs.

The Liouville transformation of a second-order equation with $t={\int }^x\sqrt{f}dx$ gives $\ddot{U}-(1+F)U=0$. Neglecting this correction yields amplitude $f^{-1/4}$ and phase or exponent $\int \sqrt{f}$. Its smallness measures a local slowly-varying-coefficient approximation, away from zeros of $f$.

Wavefunction normalization makes the total Born rule probability equal to one. In an orthonormal basis of energy eigenstates, $\psi ={\sum }_n{c}_n{\varphi }_n$ is normalized precisely when ${\sum }_n|c_n{|}^2=1$.

For a proposed infinite square well mode $A\sin (n\pi x/L)$, the integral of its squared modulus is $|A{|}^{2}L/2$. Thus wavefunction normalization requires $|A|=\sqrt{2/L}$; choosing $A$ real and positive fixes a convenient global phase. For the equal quantum superposition $({\varphi }_1+{\varphi }_2)/\sqrt{2}$, orthogonality makes the integrated cross term zero, leaving $1/2+1/2=1$. The cross term can nevertheless be nonzero locally: normalization constrains total probability, not the shape of its probability density.

Ice is solid water. Ordinary terrestrial ice has an open hydrogen-bonded crystal structure and a lower density than liquid water near the melting point.

Different pressure-temperature conditions stabilize different ice structures. Consequently, the word ice identifies a chemical substance and phase family without uniquely specifying its crystal structure.

For differentiable vector fields, the displayed identity relates vorticity to the antisymmetric part of the velocity gradient. Together with the curl of a cross product, it converts the adjoint linearized Navier-Stokes evolution between component and rotational forms.

Circular streamlines and constant vorticity imply $u_{\varphi }=\omega r/2+C/r$. Zero velocity at $2a$ and positive azimuthal velocity $V$ at $a$ giveThe radial Euler equations for an inviscid fluid give $p_r=\rho u_{\varphi }^2/r$, yielding outer-minus-inner pressure $(15-16\log 2)\rho V^2/18$. The pressure difference is independent of circulation orientation.

A Newtonian fluid has viscous stress proportional to the local rate-of-strain tensor, with coefficients independent of that rate. For an isotropic fluid these coefficients are viscosities. Its Newtonian fluid stress tensor specifies the corresponding constitutive relation; the material and its stress tensor are distinct concepts.

A fully developed parallel flow has a velocity profile independent of the streamwise coordinate. In a planar channel a representative profile is $\mathbf{u}=u(z)\hat{\mathbf{x}}$; wall conditions and translational symmetry remove cross-stream motion. The streamwise pressure gradient may remain nonzero, while the convective acceleration vanishes. Fully developed flow is a modeling branch, not a uniqueness assertion for all flows in that geometry.

Two finite isolated steady gravitational systems at the same conserved energy obey $2T+W=0$ at each endpoint, so both have $T=-E$ and $W=2E$. Redistribution can increase central concentration without making the global potential energy more negative. Escaping matter or an external reservoir requires a different energy accounting.

The virial temperature characterizes the thermal energy of gas supported in a gravitational potential. It is of order $\mu m_{p}GM/(k_{B}R)$, with a structural coefficient determined by the density profile. For a uniform self-gravitating gas sphere the virial theorem gives $k_{B}T=GM\mu m_p/(5R)$; a conventional halo circular-speed definition often uses $k_{B}T\simeq \mu m_p{v}_c^2/2$.

For positive masses in their centre of mass frame, ${\sum }_{j<k}{m}_j{m}_k|{\mathbf{r}}_j-{\mathbf{r}}_k{|}^2=M{\sum }_k{m}_k{r}_k^2$. Thus the scalar inertia $I=\frac{1}{2}{\sum }_k{m}_k{r}_k^2$ equals $(2M{)}^{-1}{\sum }_{j<k}{m}_j{m}_k{r}_{jk}^2$. The identity compares total spatial extent with pairwise separations.

For an isolated Newtonian point-mass system, $\ddot{I}=T+E\ge E>0$. Twice integrating gives a quadratic lower bound on $I$. The maximum-separation inertia bounds then give at-least-linear asymptotic growth of the diameter $R$ for a trajectory existing for arbitrarily large times. The bound does not assert instantaneous monotonicity or escape of every particle.

The vertical velocity is the component of a fluid velocity field along the selected upward vertical direction. In incompressible flow, impermeable horizontal boundaries impose $w=0$. Eliminating pressure often yields a closed hydrodynamic stability equation for this component.

Set-valued analysis studies relations whose output at one input is a set. Solution maps, subdifferentials and generalized normal mappings are important examples. Graph geometry provides definitions of continuity, local stability and differentiation.

Normal cones describe generalized supporting directions to sets. For a convex set, the normal condition is $⟨w,z^{\prime }-z⟩\le 0$ for every $z^{\prime }$ in the set. For nonconvex sets, the Fréchet normal cone and the limiting normal cone distinguish local regular support from limits of such supports.

Writing $f=1-2M(v)/r$, the radial Christoffel symbols are ${\Gamma }^v{}_{vv}=M/r^2$, ${\Gamma }^r{}_{vv}=fM/r^2+M^{\prime }/r$ and ${\Gamma }^r{}_{vr}=-M/r^2$. The angular couplings are ${\Gamma }^v{}_{\theta \theta }=-r$, ${\Gamma }^r{}_{\theta \theta }=-fr$, with ${\sin }^2\theta$ factors for the corresponding $\varphi \varphi$ terms, ${\Gamma }^{\theta }{}_{r\theta }={\Gamma }^{\varphi }{}_{r\varphi }=1/r$, ${\Gamma }^{\theta }{}_{\varphi \varphi }=-\sin \theta \cos \theta$ and ${\Gamma }^{\varphi }{}_{\theta \varphi }=\cot \theta$. The lower indices are symmetric. The plus sign of $M^{\prime }/r$ follows from the ingoing $+2dvdr$ cross term.

The two radial null geodesic families are $v=\text{constant}$ and $dr/dv=(1-2M(v)/r)/2$. The first has affine radius, while the second has tangent $k={\partial }_v+(f/2){\partial }_r$ obeying ${\nabla }_{k}k=(M/r^2)k$, so advanced time is generally a nonaffine parameter. An affine parameter $\lambda$ for the outgoing family satisfies $d\log (dv/d\lambda )/dv=-M/r^2$.