An effective gas pressure defined by $\mu ={\mu }^{\circ }+RT\ln (f/p^{\circ })$ in a specified standard-state convention. It approaches the gas pressure at low density and differs from the dimensionless fugacity.

A synapsid in the lineage that includes cynodonts and mammals. Many therapsids lie outside the mammalian crown group.

The ideal fields after instantaneous switching behave as $E_z,H_{\varphi }\propto (t-r/c{)}^{-1/2}$ near arrival, and the radial Poynting vector as $(t-r/c{)}^{-1}$. A finite switching rise time smooths the ideal front; the singular solution does not represent a finite-energy real pulse.

The induced electric field opposes the increasing axial current, while the magnetic field circles it. Their Poynting vector points radially outward and transports energy into the growing magnetic-field region.

Parametrize the ellipsoid by $\mathbf{x}=(b\sin \theta \cos \varphi ,b\sin \theta \sin \varphi ,a\cos \theta )$, with positive semiaxes $a,b$, $0\le \theta \le \pi$ and $0\le \varphi <2\pi$. The cross product ${\mathbf{x}}_{\theta }\times {\mathbf{x}}_{\varphi }$ gives the displayed outward vector area element. For $0<\theta <\pi$, it is the positive multiple $a{b}^2\sin \theta$ of the outward normal direction $(x/b^2,y/b^2,z/a^2)$, fixing its orientation. A flux integral can therefore be evaluated directly as $\int \int \mathbf{F}(\mathbf{x})\cdot ({\mathbf{x}}_{\theta }\times {\mathbf{x}}_{\varphi })d\theta d\varphi$, without computing the length of the normal separately. The scalar surface area element is the Euclidean norm of this vector element.

On an origin-centered radius-$a$ sphere, $\mathbf{n}=\mathbf{x}/a$. Spherical symmetry gives ${\int }_{S_a}{x}_{i}dS=0$ and ${\int }_{S_a}{x}_i{x}_{j}dS=(4\pi a^4/3){\delta }_{ij}$: off-diagonal moments vanish by reflection, diagonal moments are equal, and their sum is $a^2$ times the surface area. Contracting these identities with $M_{ij}/a$ proves the displayed flux directly, including a zero contribution from a constant field $\mathbf{b}$. It agrees with the divergence theorem, since $\nabla \cdot (M\mathbf{x}+\mathbf{b})=\mathrm{tr}M$.

Take the upward-oriented graph $z=c\sqrt{1-x^2/a^2-y^2/b^2}$ over $0\le x\le L_x$, $0\le y\le L_y$, where $a,b,c,L_x,L_y>0$, $A=L_x^2/a^2$, $B=L_y^2/b^2$ and $A+B<1$. The surface flux of the vector field $F=(-y,x,0)$ is the displayed formula. Indeed the vector surface element is $(-z_x,-z_y,1)dxdy$, soThe change of variables $u=x^2/a^2$, $v=y^2/b^2$ changes $xydxdy$ to $a^2{b}^{2}dudv/4$. Integrating $(1-u-v{)}^{-1/2}$ over $[0,A]\times [0,B]$ gives $\frac{4}{3}[1-(1-A{)}^{3/2}-(1-B{)}^{3/2}+(1-A-B{)}^{3/2}]$, proving the result. When $a=b$, the value of the vector field is a tangent vector to the ellipsoid at every point, and the surface flux vanishes pointwise.

On the graph of a function $z=f(x,y)$, assume $f$ is differentiable, its planar gradient $p$ is nonzero, and speed $U>0$ is measured by the three-dimensional Euclidean norm. The upward unit normal vector is $n=(-p,1)/\sqrt{1+|p{|}^2}$. Projecting the downward vertical vector $-e_z$ onto the tangent plane gives $-(p,|p{|}^2)/(1+|p{|}^2)$. By the Cauchy-Schwarz inequality, normalizing this projected vector produces the tangent direction with greatest height decrease at the prescribed speed, giving the displayed velocity. Its vertical component is $-U|p|/\sqrt{1+|p{|}^2}$, and its horizontal direction is parallel to $-\nabla f$. Consequently its planar paths agree with negative gradient flow up to a change of time, although the speeds differ. Where $p=0$, every tangent direction has zero first-order height change and this normalized formula is undefined.

Let ${\nabla }_{\tan }F=\nabla F-{\mathbf{e}}_r{\partial }_{r}F$ be the tangential surface gradient on a sphere of radius $r>0$. For smooth scalar fields, set $\mathbf{f}=R{\mathbf{e}}_r+{\nabla }_{\tan }G+{\mathbf{e}}_r\times {\nabla }_{\tan }F$. Its radial curl detects only $F$:To prove this, use the radial curl formula $(r\sin \theta {)}^{-1}[{\partial }_{\theta }(\sin \theta f_{\varphi })-{\partial }_{\varphi }{f}_{\theta }]$. The radial component $R$ never enters it. The $G$ terms cancel as $G_{\varphi \theta }-G_{\theta \varphi }=0$ by symmetry of second derivatives, while the $F$ terms give the unit-sphere Laplace-Beltrami operator. Taking the dot product with the position vector $\mathbf{r}=r{\mathbf{e}}_r$ gives the displayed $r^{-1}$ version. This tangential gradient decomposition does not assert that the whole vector field is solenoidal.

The fractional excess of water-vapor pressure above equilibrium saturation over a flat pure-water surface at the same temperature: $s=p_v/p_{\mathrm{sat}}(T)-1$. Positive $s$ can drive droplet growth, but activation of a particular aerosol particle requires exceeding its critical supersaturation. Cooling, condensation and mixing jointly determine parcel supersaturation.

2016 paper: Public original PDF, physical page 839. Student-answer availability for this paper remains unverified.

A finite set $A$ of positive integers such that nonempty disjoint sets contained in $A$ have different sums, and the larger subset has the larger sum when their sizes differ. For $a_1<\cdots <a_n$, the size condition is equivalent to $a_1+\cdots +a_{k+1}>a_{n-k+1}+\cdots +a_n$ for $1\le k\le \lfloor (n-1)/2\rfloor$: the left side is the smallest possible sum of $k+1$ elements and the right side the largest sum of $k$ elements.

For $\epsilon =\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$, every two-by-two matrix satisfies $U^T\epsilon U=(\det U)\epsilon$. Thus ${\varphi }^T\epsilon \chi$ is exactly invariant under a common SU(2) group transformation. This is a bilinear contraction without complex conjugation. It vanishes for equal commuting spinors and is the alternating tensor defining the spin singlet state. Overall bosonic exchange symmetry must still be enforced if the spinors label identical particles.

Cold-region stratospheric particles that can support heterogeneous reactions activating reactive halogen chemistry. Their occurrence and effect depend on temperature, available condensable species, halogen reservoirs and sunlight history.

An increase of atmospheric carbon dioxide can cool the stratosphere through enhanced infrared cooling relative to non-infrared heating and through changes in the spectrum of upwelling radiation. A local emissivity-only explanation is incomplete because absorption and emission both change. The response depends on ozone solar heating, spectral opacity, altitude and adjustments of the layers below.^[ref]

A wintertime region of strong circumpolar flow that can isolate polar stratospheric air and sustain cold conditions. Isolation, cloud formation, sunlight return and subsequent breakup jointly influence seasonal ozone chemistry. Cooling alone does not determine the full ozone evolution.

The wave-driven mean stratospheric circulation with tropical ascent and extratropical descent. Changes in wave propagation and momentum deposition can alter its strength and transport times. Its response redistributes ozone and trace gases; local chemistry and transport must be considered together.

Transport of air and trace species across the tropopause. Downward ozone transport can supply the troposphere even when local tropospheric photochemical production is unchanged. Regional pathways, mixing and residence times determine the resulting burden.

For dilute isolated spheres in a Newtonian solvent, $D\eta a/T=k_B/(6\pi )$. An experimental check independently measures temperature, viscosity and particle size, then obtains diffusion from a displacement variance or a scattering decay. Drift, walls, concentration effects, aggregation and a hydrodynamic radius different from a structural radius can defeat an unqualified comparison.

A right-handed neutrino can transform as $(1,1,0)$ under colour, weak isospin and hypercharge. It has no direct Standard Model gauge coupling, but may mix with active neutrinos. Sterility under the low-energy group does not imply sterility under an enlarged unbroken gauge group.