On an oriented $d$-dimensional pseudo-Riemannian manifold, the metric volume tensor is the totally antisymmetric tensor with ${ϵ}_{1\dots d}=\sqrt{|\det g|}$ in positively oriented coordinates. It supplies the components of the volume form. It differs from the coordinate Levi-Civita symbol, whose entries are just $0,1,-1$. Raising indices with a Lorentzian metric can change signs; the metric, dimension and orientation must be specified.

The oriented metric volume form on a two-dimensional surface. On the unit sphere with standard orientation, $\omega =\sin \theta d\theta \wedge d\phi$ and $\int \omega =4\pi$. Its pullback of a differential form under a map measures signed local area.

On an oriented Riemannian manifold, the metric volume form is the volume form taking value one on positively oriented orthonormal frames. In an oriented manifold chart it has the displayed expression. Under an orientation-preserving coordinate change with Jacobian $A$, $\det g$ acquires the factor $(\det A{)}^2$, precisely matching the transformation of the top exterior product. Thus the expressions agree on chart overlaps. Orientation is needed to make a globally positive form, though the volume density exists without it.

If $\Vert \tau f(t)\Vert \le C{\int }_0^t\Vert f(s)\Vert ds$, repeated integration produces the time-ordered simplex of volume $t^n/n!$. Thus the Neumann series ${\sum }_{n\ge 0}{\tau }^n$ converges on every finite interval, without requiring the single-step operator norm to be less than one.

Starting from $\Vert \tau g(t){\Vert }^2\le C^{2}t{\int }_0^t\Vert g(s){\Vert }^{2}ds$, induction integrates $s^{2n-2}$ at the $n$th step. Its denominator accumulates the odd integers $1,3,\dots ,2n-1$. The factorial bound for a Volterra iterate is stronger because $(n!{)}^2\ge (2n-1)!!$.

For positive diffusivity, the Cole-Hopf transformation with $f=2\alpha {\partial }_{\theta }\log \psi$ converts the negative-flux equation into the heat equation. Step data give a ratio of two Gaussian-tail integrals $J$. At $\theta =-UZ/2$, $J=1$ and $f=U/2$. Its vanishing-viscosity limit selects the shock or rarefaction wave in the Burgers Riemann problem with negative flux, depending on the sign of $U$.

A localized negative-flux Burgers N-wave has initial profile $-U\theta$ on $|\theta |<L$ and zero outside, for $U,L>0$. The Cole-Hopf solution for a Burgers N-wave expresses its viscous evolution in Gaussian interval masses. In the vanishing-viscosity limit, the interior slope becomes $-U/(1+UZ)$ and its entropy-shock fronts move to $\pm L\sqrt{1+UZ}$.

Combining a virial-temperature relation with the density of a uniform self-gravitating sphere gives $M\propto T^{3/2}{n}_H^{-1/2}$. Thus contours of constant mass in a density-temperature plane satisfy $n_H\propto T^3/M^2$. Structural coefficients, composition and the distinction between baryonic and total gravitating mass matter when giving numerical labels.

A vector space with scalar field $\mathbb{Q}$ is equivalently a torsion-free divisible Abelian group. An embedding of such groups automatically respects rational scalar multiplication because solutions to $nb=ma$ are unique.

For the metric in vector cosmological perturbation, an orthonormal tetrad is $E_0=a^{-1}{\partial }_{\eta }$, $E_i=a^{-1}(B_i{\partial }_{\eta }+{\partial }_i)$ to first order. A photon has $p^0=(ϵ/a^2)(1+B_i{e}^i)$ and $p^i=(ϵ/a^2)e^i$, so the covariant component is $p_0=-ϵ+O(B^2)$. The covariant geodesic equation gives $d{p}_0/d\eta =(2p^0{)}^{-1}{\partial }_{\eta }{g}_{\alpha \beta }{p}^{\alpha }{p}^{\beta }=ϵe^i{\dot{B}}_i$. The derivative of the scale factor cancels by the null vector condition, proving the redshift formula.

For a holomorphic vector bundle $E$, this sheaf assigns to $U$ the smooth $E$-valued differential forms of type (p, q) on $U$. Its Dolbeault operator acts coefficientwise in a holomorphic local frame and squares to zero.

For a polynomial over a discretely valued field and a rational weight $t$, this coefficient-minimum function is multiplicative. Scale the variable by an element of valuation $t$ in a valued extension and reduce the initial terms; nonzero residue polynomials have nonzero product. For a monic polynomial factored into roots, $w_t(f)={\sum }_r\min (t,v(r))$. Its supporting lines determine the lower Newton polygon, giving the Newton polygon root valuation theorem and its multiplicities.

A nontrivial additive valuation has rank one when its ordered value group is Archimedean and therefore embeds order-preservingly in $\mathbb{R}$. Exponentiating a real-valued representative gives a Non-Archimedean absolute value; changing the embedding rescales the valuation positively. Higher-rank ordered value groups cannot be represented by real-valued absolute values on a field.

A spatially homogeneous field value minimizing the potential energy. For a finite-energy field configuration on an infinite domain, the potential is normally shifted so the relevant vacuum energy density is zero. Disconnected components of the vacuum manifold allow a scalar-field kink to connect distinct vacuum values.

A vacuum spacetime has vanishing matter stress-energy tensor. In $D>2$ dimensions, tracing the Vacuum Einstein equations gives $R=2D\Lambda /(D-2)$ and hence $R_{ab}=2\Lambda g_{ab}/(D-2)$; in four dimensions this reduces to $R_{ab}=\Lambda g_{ab}$. With zero cosmological constant it is Ricci flat, although its full Riemann curvature tensor can be nonzero: a vacuum Kerr black hole and vacuum gravitational waves illustrate that distinction.

Risk neutrality means evaluating uncertain monetary gains by their expected value. For auction payment identities, quasilinear utility gives a bidder's expected utility as its value times its winning probability minus expected payment.

This utility is linear in the monetary transfer: allocation value minus payment. It supports the incentive comparisons and payment identities used in mechanism design.

A constant relative risk aversion utility has $-c{U}^{\prime \prime }(c)/U^{\prime }(c)=R>0$. Up to a positive scale and an additive constant, it is $U(c)=c^{1-R}/(1-R)$ for $R\ne 1$ and $U(c)=\log c$ for $R=1$. Power utility gives multiplicative homogeneity of an investment value; logarithmic utility gives additive scaling.

The Inada conditions specify infinite marginal utility at zero consumption or wealth and zero marginal utility at infinity. For an increasing strictly concave utility function, they make the inverse marginal utility map cover all positive shadow prices, facilitating the state-price budget constraint method.

Multiplicative habit utility modifies current consumption utility through its ratio to habit $h>0$. With $U(c)=c^{1-R}/(1-R)$, $R>1$, and $\alpha >0$, it is $h^{\alpha }{c}^{1-R-\alpha }/(1-R)$. Its homogeneity in $(c,h)$ has degree $1-R$, while consumption holding habit fixed has effective relative risk aversion $R+\alpha$.