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The derivative of internal energy at fixed deformation gradient and entropy, or of specific Helmholtz free energy at fixed temperature, defines this force. Its product with the internal-variable rate contributes mechanical dissipation to the Clausius-Duhem inequality.

Thermodynamics studies energy transfers and the constraints on macroscopic changes. Its vocabulary connects internal energy, entropy, temperature, pressure and volume to heat and work. The first law of thermodynamics balances energy; the Second law of thermodynamics decides which changes can occur spontaneously and limits how much work a heat engine can deliver. These principles let us bound performance before choosing a material or a detailed machine design. Applications include power generation, propulsion and refrigerators. ^[ref]

Start by choosing a system boundary. A closed system can exchange energy but no matter; an isolated system exchanges neither. A state function, such as internal energy or entropy, depends on the equilibrium state. A process specifies how that state changes. Thus a body's temperature is a property it has, whereas heat records energy crossing its boundary during a process. A body contains internal energy, rather than a stored quantity of heat.

Two insulated bodies initially at different positive temperatures approach thermal equilibrium when brought into thermal contact. Their combined energy is conserved, but their combined entropy increases. They do not spontaneously separate again into a hotter body and a colder one. This familiar asymmetry makes thermodynamics interesting: conserving energy leaves many imaginable changes available, and entropy supplies a further constraint. The same distinction underlies both entropy maximization under thermal contact and the thermal efficiency limit in Carnot's theorem.

A thermal distribution of relative velocities weights a nuclear cross-section to determine the reaction rate per volume. Identical reactants require the denominator two to avoid counting each pair twice. Specific deposited power is $r_{12}{Q}_{\mathrm{deposited}}/\rho$. Charged-particle quantum tunnelling through a Coulomb barrier produces a Gamow peak, while resonances and plasma screening can alter the rate.

Thermoregulation regulates body temperature. Skin signals can anticipate a change of core body temperature, so a response need not wait for an internal error. ^[ref]

These responses have costs and environmental limits. Shivering consumes metabolic resources; cooling through sweating removes water that must eventually be replaced. High humidity restricts evaporation, so secreted sweat and successful cooling are not interchangeable. Maintaining core body temperature also depends on maintaining circulation: depleted blood volume can compromise skin blood flow and heat transfer. Seeking shade or reducing work changes the disturbance itself, whereas a larger regulatory command merely attempts to counter it. ^[ref]

In the Alexander-de Gennes brush model, a theta solvent gives $D\sim a{g}^{1/2}$. Stacking $N/g$ blobs then gives the displayed height, with mean polymer volume fraction $\varphi \sim (s{a}^2{)}^{1/2}$ for segment volume $a^3$. Cancellation of the two-segment excluded volume does not remove three-segment repulsion or make the crowded layer freely interpenetrable at no cost.

A chemist.^[ref]

An arterial reservoir, venous reservoir and a left ventricle exchange flows $Q_a,Q_v,Q_c$. Their volume balances are $C_a{\dot{p}}_a=Q_a-Q_c$, $C_v{\dot{p}}_v=Q_c-Q_v$, and $d(p_{LV}/E_{LV})/dt=Q_v-Q_a$. Ideal one-way valves give $Q_a=[p_{LV}-p_a{]}_{+}/R_a$, $Q_v=[p_v-p_{LV}{]}_{+}/R_v$; systemic resistance gives $Q_c=(p_a-p_v)/R_c$. Summing the three equations verifies total mass conservation for incompressible blood.

For the uniform noninteracting electron model with two spin states and kinetic energy of nonrelativistic particles $E=p^2/(2m_e)$, count all available one-particle states up to an energy $E\ge 0$:This extensive density of states has units of inverse energy; $g(E)/V$ is the density of states per unit volume. Availability and occupation are different: $g(E)$ is positive above $E_F$, but at zero temperature those states are empty. The occupied energy distribution is $g(E)f(E)$, with the Fermi-Dirac distribution reduced to $f(E)=1$ below $E_F$ and $f(E)=0$ above it. The convention at the exact boundary does not affect a continuum integral.

With the dimensionless quantity $x=E/E_F$, the occupied count per $dx$, divided by the total particle number, is $3\sqrt{x}/2$ for $0<x<1$. Its area is one. Weighting this by $E$ gives the internal energy and its mean per particle:Thus the highest occupied energy and the mean energy differ. For the Fermi energy of a model electron gas, $U/N\simeq 1.0155\mathrm{eV}$. At fixed $N$, $E_F\propto V^{-2/3}$, so the derivative $P=-(\partial U/\partial V{)}_N$ gives the electron degeneracy pressure $P=2n{E}_F/5$. The same state-counting geometry therefore predicts a nonzero pressure at zero temperature.

For isotropic direct-gap parabolic bands, $m_r^{-1}=m_e^{\ast -1}+m_h^{\ast -1}$ and transition energy is $E_g+{\hslash }^2{k}^2/(2m_r)$. Differentiating the two-spin state count $k^3/(3{\pi }^2)$ per volume gives the displayed density for $E>E_g$, and zero below. Density per angular frequency is $\hslash g_j(\hslash \omega )$, not the same dimensioned quantity.

A point injection of volume at rate $Q$ produces the displayed radial potential flow in three dimensions. Its velocity potential is $-Q/(4\pi r)$, and every enclosing sphere has outward volume flux $Q$. The divergence vanishes away from the source and is $Q$ times the three-dimensional Dirac delta distribution at it. The field is also a Stokes flow with constant pressure away from its singularity.

Aligned two-dimensional sections constrain the three-dimensional structure of a specimen. For parallel sections of known spacing $\Delta z$, the Riemann sum ${\sum }_i{A}_i\Delta z$ approximates volume. Distortion, missing sections and alignment errors constrain the reconstruction; the volume estimate alone does not determine the identity or developmental fate of each visible structure.

Tonicity describes how a surrounding chemical solution changes cellular volume through osmosis. It depends on which solutes remain effectively excluded on the timescale of interest, not simply on the amount of dissolved material. The following comparisons take a functional cell membrane, an initially unpressurized biological cell, and effectively excluded solutes as the reference; rising turgor pressure can oppose entry in a walled biological cell.

Operationally, hypotonic solutions cause sustained swelling, hypertonic solutions sustained shrinking, and isotonic solutions no sustained volume change in the specified passive comparison. State the reference biological cell, membrane properties and observation interval. A permeating solute can transiently change water potential while crossing; after its chemical potential equilibrates, its initial unequal concentrations cannot be treated as a permanently retained osmotic load. Thus equal osmolarities need not imply isotonic solutions. Conversely, a stationary walled biological cell may have unequal osmolarities balanced by turgor pressure.^[ref]

In the steady small-Rossby number transport equation, this coefficient field advects the ocean transport streamfunction along contours of background shallow-water potential vorticity $f/H$. It is not a physical velocity: when $\psi$ has units of volume transport, its units are inverse length squared per time. For uniform depth on a beta plane it points westward, $\tilde{\mathbf{u}}=(-\beta /H,0)$. With no forcing or drag, $\psi$ is constant on each connected characteristic; locally $\psi =\Phi (f/H)$ at regular points.

For a smooth Skyrme model field $U:{\mathbb{R}}^3\to SU(2)$ with $U(\infty )=1$, compactification gives a map $S^3\to S^3$. With $L_i=U^{†}{\partial }_{i}U$, its degree of a map between oriented manifolds is $B=-(24{\pi }^2{)}^{-1}\int {ϵ}_{ijk}\mathrm{tr}(L_i{L}_j{L}_k)d^{3}x$, with sign chosen so the standard decreasing hedgehog profile has $B=1$. The integrand is the pullback of the normalized volume form on $SU(2)$. Hence it is integer-valued and unchanged by smooth finite-energy homotopies. In the rational map approximation for Skyrmions, a degree-$N$ angular map and profile $f(0)=\pi ,f(\infty )=0$ give $B=-(2N/\pi )\int f^{\prime }{\sin }^{2}fdr=N$. This is a topological conservation law, not the Noether charge of isospin.

A local current whose conservation follows from field geometry rather than requiring the field equation. For Skyrmions, pulling back the closed volume form on the target three-sphere gives the current associated with topological baryon number in the Skyrme model. Its spatial integral is a topological charge, unchanged by smooth evolution with the prescribed boundary behaviour.

A toroid is a ring-shaped surface of revolution obtained by rotating a closed planar profile about an axis in its plane that stays outside the profile. A circular profile gives a torus; a rectangular profile gives a ring with rectangular cross-section. The enclosed solid supplies the core geometry in a toroidal winding, while the mathematical surface is its boundary. The distinction matters when computing a surface area versus a volume.

An Abelian K-matrix Chern--Simons theory has $|\det K|$ topologically distinct ground states on a torus.

For a noncompact free boson with target volume $V$, the torus partition function is $Z=V[2\pi \sqrt{\mathrm{Im}\tau }{]}^{-1}|\eta (\tau ){|}^{-2}$ in the normalization where the zero-mode Hamiltonian is $p^2/2$.