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Every isometry of a complete singular flat cone, of cone angle $\theta \ne 2\pi$, fixes its vertex: this is the unique point without a locally Euclidean neighbourhood. It preserves distance from that vertex and has the formThese are angular rotations and reflections. The stabilizer of a nonvertex point consists of the identity and its radial reflection. The vertex stabilizer is the entire isometry group, including all rotations: an unspecified fixed point does not give the two-element conclusion.

When $\theta =2\pi$, the complete cone is the Euclidean plane, allowing translations and every rotation about a fixed point. This smooth case is excluded from the singular-cone statement.

For a connected flat Riemannian surface $S$, a developing map is a local isometry $\mathrm{dev}:\tilde{S}\to {\mathbb{R}}^2$ from its universal cover into the Euclidean plane. Continuing local Euclidean coordinates along paths constructs it, up to a Euclidean motion determined by the initial chart. It need not be injective or cover the entire plane.

Chart continuation around loops defines a holonomy representation $\rho :{\pi }_1(S)\to \mathrm{Isom}({\mathbb{R}}^2)$ satisfyingfor deck transformations $g$. Unrolling a circular cylinder gives a developing map whose generating deck transformation becomes translation by the circumference. On a punctured flat cone, lifted angular coordinates give $\mathrm{dev}(r,t)=(r\cos t,r\sin t)$. A circuit sends $t$ to $t+\theta$, giving rotation by the cone angle. This describes how flat charts fit together, despite vanishing local Gaussian curvature.

The physical tension is $f=k_{B}T\tau$. At small extension,At fixed tension the extension is $L=na\tanh (fa/(k_{B}T))$ and decreases as temperature rises.

For continuous $f:[0,1]\to [0,1]$, the function $f(x)-x$ is nonnegative at zero and nonpositive at one, so the intermediate value theorem gives a fixed point.

If $h:A\to B$ is a homeomorphism and $A$ has the fixed-point property, then a continuous $f:B\to B$ induces $h^{-1}fh:A\to A$. A fixed point of the conjugate map is carried by $h$ to a fixed point of $f$.

The Löwenheim--Skolem theorems control the cardinalities of models of a first-order theory without changing its first-order truths.

A set of first-order sentences has a model if and only if every finite subset has a model.

Every sentence derivable from a first-order theory is true in every model of that theory.

This is soundness of the first-order deduction system relative to its structure semantics.

A structure supplies a nonempty domain and an interpretation of every constant, operation, and relation symbol in its language.

This is a structure for a first-order language, with finitary interpretations of its nonlogical symbols.

A first-order theory is a set of sentences in one first-order language. Depending on convention, the term may instead mean its deductive closure.

This is a theory formulated in first-order logic.

A first-order sentence is a formula with no free variables, so its truth in a structure is independent of a variable assignment.

A sentence has no free variables; a first-order sentence is its first-order instance.

An occurrence of a variable is free when it is outside the scope of a quantifier binding that variable.

The distinction belongs to free variables and bound variables; an occurrence bound by an enclosing quantifier is not free.

A first-order language specifies its constant, operation, and relation symbols, together with the arity of each nonconstant symbol.

A first-order formula is built from atomic formulas using logical connectives and quantifiers over the elements of a structure.

An adiabatic process has no heat transfer across the chosen boundary throughout the process: $\delta Q=0$, rather than merely zero net heat after incoming and outgoing transfers cancel. It is a condition on energy transfer, whereas an isothermal process holds temperature constant. A sealed cylinder can approach an adiabatic process through insulation or through motion short compared with the time needed for appreciable heat transfer. Its temperature can nevertheless change because work crosses the boundary.^[ref]

Take the trapped gas as a closed system, with fixed amount of substance and composition and no net change in bulk kinetic energy or potential energy between the endpoints. Define $Q_{\mathrm{in}}>0$ as heat entering and $W_{\mathrm{out}}>0$ as work delivered to the surroundings. The first law of thermodynamics givesHere $U$ is internal energy. An adiabatic compression normally supplies work, so $W_{\mathrm{out}}<0$; an adiabatic expansion against a resisting load delivers work, so $W_{\mathrm{out}}>0$. An adiabatic process need not preserve internal energy and need not make the closed system an isolated system. ^[ref]

For a reversible thermodynamic process, $dS=\delta Q_{\mathrm{rev}}/T$ at positive absolute temperature $T$. A reversible adiabatic process therefore has $dS=0$: it is an isentropic process. Irreversible adiabatic processes can instead generate entropy; free expansion is the existing counterexample, with its calculation in entropy change in free expansion. Conversely, constant entropy alone does not certify an adiabatic process or reversibility: outward heat can export the entropy generated internally. For a closed system the macroscopic entropy balance iswhere $T_b$ is the absolute boundary temperature where each heat transfer occurs and $S_{\mathrm{gen}}$ is internal entropy production. With no heat transfer, this reduces to $\Delta S=S_{\mathrm{gen}}$; reversibility makes $S_{\mathrm{gen}}=0$. ^[ref]

A motivating experiment measures the gas temperature directly during compression and expansion, instead of inferring it from how warm a cylinder feels. PASCO's authored TD-8565 manual describes a sealed piston cylinder, a fine nickel-wire thermometer, an absolute pressure sensor and a position readout for volume. A computer records all three channels together. The wire's electrical resistance changes with temperature; its small size permits a rapid response. ^[ref]

Use ordinary trapped air and the apparatus's designed travel stops; mount the base securely, close the valve during a stroke, and record the actual volume limits. John Hart's university demonstration uses a large transparent cylinder with a low-heat capacity thermocouple, a pressure sensor, a travel gauge and a safety stop. It explicitly keeps the release valve closed during the demonstration. The basic observable is a rise in temperature on compression and a fall on loaded expansion, alongside the corresponding pressure and volume changes. Hart discusses wall thermal conduction, sensor response and leakage as reasons real records differ from the reversible model. ^[ref]

For a proposed rapid-versus-slow comparison, let the air and walls return to the same initial thermal equilibrium before each run, then compare the same starting and finishing volume values with different stroke durations. Read temperature, absolute pressure and position continuously, including the hold after motion stops. A rapid compression gives a temporary temperature rise; holding the smaller volume lets heat leave and the temperature approach the surroundings again. Slow compression in good thermal contact allows that heat transfer during the stroke and approaches an isothermal compression. Loaded expansion reverses the signs: the rapid stroke cools, while heat entering during slow motion reduces the cooling. PASCO explicitly supports rapid nearly adiabatic processes and slow isothermal processes; the matched-run controls here are a proposed extension, not a reported set of measurements. ^[ref]

Insulation suppresses thermal conduction; it does not prove that heat transfer vanishes. As proposed controls, repeat the matched strokes with and without a removable insulating jacket, record room and wall temperatures, measure fixed-position drift, and check valve closure and seal leakage. Calibrate the thermometer and position readout independently, including any connected dead volume; account for the heat capacity of the probe rather than treating it as a passive observer. Hart explicitly identifies the small air sample relative to the probe and walls, sensor inertia and seal leakage as limitations. A fast display cannot eliminate those physical effects. ^[ref]

There are two distinct timing requirements. Let ${\tau }_{\mathrm{eq}}$ describe internal equilibration, ${\tau }_{\mathrm{probe}}$ the thermometer response, $t_{\mathrm{move}}$ the stroke duration and ${\tau }_{\mathrm{heat}}$ the effective heat transfer time constant. A useful sufficient window for a resolved, nearly reversible adiabatic process iswith negligible friction. The left inequality permits a quasistatic process and a usable readout; the right limits heat leakage. A violent stroke may satisfy only the right inequality. PASCO reports a roughly $30$–$50\mathrm{ms}$ lag of its temperature channel behind the other channels, so apparent simultaneous samples require care. If no such window exists, analyze equilibrium endpoints with the first law of thermodynamics, and regard the path as approximate. ^[ref]

The diagram's model is deliberately a hold after the stroke, when boundary work is zero. For constant whole-sample heat capacity at constant volume $C_V$ and effective thermal conductance $K>0$, it assumesThis is Newton's law of cooling with one uniform gas temperature and a fixed surrounding temperature; actual wall and probe dynamics can require more than one time constant. At $t/{\tau }_{\mathrm{heat}}=0.1$, the fraction of the initial temperature anomaly erased is $1-e^{-0.1}\simeq 0.0952$; at one time constant it is about $0.6321$. These are analytical fractions, not experimental tolerances or evidence that the preceding stroke was reversible. Measuring the hold helps estimate heat leakage independently of the motion.

The usefulness of an adiabatic process is that it separates mechanical work from heat transfer. Its reversible limit connects temperature changes, mechanical stiffness and entropy in one model: Rüchardt experiment and Clément-Desormes experiment already use that connection to measure heat capacity ratio. The contrast with free expansion explains the deeper point: suppressing heat does not determine how much work is exchanged or how much entropy is produced.

A Riemannian metric on a smooth surface defines an unsigned area density of a surface in every chart:where $E,F,G$ are the coefficients of the first fundamental form. For a coordinate change with Jacobian matrix $J$, the identity $\tilde{g}=J^{T}gJ$ gives $\sqrt{\det \tilde{g}}=|\det J|\sqrt{\det g}$. This is exactly the absolute Jacobian determinant factor in coordinate area substitution, so the density defines surface area globally, including on a nonorientable surface.

If an orientation of a surface is also chosen, the corresponding oriented area two-form is, in every positively oriented chart,Changing the orientation negates this differential form, while leaving the unsigned surface area unchanged. A Möbius band has an area density but no globally nonvanishing area two-form, since such a form would orient it. The notation $dA$ is often used for the density too; an integral computing unsigned surface area uses the positive density convention.

A local isometry between Riemannian manifolds is a local diffeomorphism $f$ with $f^{\ast }{g}_N=g_M$. Equivalently, its differentials are isometric linear isomorphisms between tangent spaces. It preserves the Levi-Civita connection, intrinsic sectional curvature, and affinely parametrized geodesics. A map whose differentials are merely isometric injections is an isometric immersion; the local diffeomorphism requirement distinguishes these notions.

Both $z$ and $p_z$ commute with $L_z$ and are odd under parity. Their angular selection rules therefore require the same magnetic quantum number and opposite orbital parity. Within the nonrelativistic point-Coulomb $n=2$ hydrogen atom eigenspace, the only angularly permitted off-diagonal pair is $|2s⟩$ and $|2p,m=0⟩$.

Those necessary angular conditions do not guarantee a nonzero matrix element. For the Coulomb potential Hamiltonian $H={\mathbf{p}}^2/(2m)+V(r)$,All states within this exact $n=2$ eigenspace have the same energy, so every projected $p_z$ matrix element vanishes. In contrast, the position matrix element is nonzero: with conventional real spherical harmonics and hydrogen radial wavefunction in the first excited shell, $⟨2s|z|2p,0⟩=-3a_0$. Its sign depends on basis phases, while its magnitude does not.

Thus the electric-dipole selection rules for hydrogen allow a linear position perturbation to mix these degenerate states, but a momentum perturbation cannot be assigned the same matrix merely from parity. The conclusion concerns the unperturbed Coulomb eigenspace; degeneracy-breaking interactions require their own energy eigenstates and matrix elements.

A perturbation $g{L}^2$ preserves the hydrogen eigenbasis and shifts $|n,\ell ,m⟩$ byIn the first excited eigenspace, the $2s$ state is unshifted and the three $2p$ states receive the common shift $2g{\hslash }^2$.

The Klein four-group is the abelian group with three nonidentity elements, all of order two.