A thermometer that infers temperature from a calibrated electrical resistance. Measuring current can heat its sensing element; extrapolation toward zero current controls that bias in the saturation-pressure equilibrium measurement.

Linearize $\det F=h(\theta )$ at $F=I$, $\theta ={\theta }_0$, $h({\theta }_0)=1$, and zero reaction. With displacement gradient $H$ and temperature increment $\vartheta$, one has $\mathrm{tr}H=h^{\prime }({\theta }_0)\vartheta$ andwhere $C_{jilk}={\rho }_0{\psi }_{,F_{ij}{F}_{kl}}$, ${\beta }_{ji}={\rho }_0{\psi }_{,F_{ij}\theta }$ and $C_e=-{\rho }_0{\psi }_{,\theta \theta }$ at the reference state. The temperature factor converts this entropy coefficient to a volumetric heat capacity: ${\theta }_0{C}_e$. Strictly constant volume is the case $h^{\prime }=0$.

Mechanical equilibrium requires force balance for the mechanical displacements that are allowed. Two uniform fluids separated by a freely moving, unloaded, frictionless piston with equal face areas require equal pressures. A supported piston can sustain a pressure difference, and a fluid in gravity instead obeys hydrostatic equilibrium. For an upward height coordinate $z$, that balance is $dp/dz=-\rho g$, where $\rho$ is mass density and $g$ is the downward gravitational acceleration. Mechanical equilibrium alone does not establish thermal equilibrium or chemical equilibrium.

Thermocouple voltage calibration relates an actual assembled sensor's electromotive force to independently established junction temperatures. It must be distinguished from selecting a nominal letter-type conversion curve. In Thermocouples Calibrations Services by National Institute of Standards and Technology (opening method description and bath-calibration table), NIST describes comparison with a maintained reference thermocouple, comparison with a standard platinum resistance thermometer in controlled heat baths, and fixed-point calibration at the freezing points of zinc, aluminum, silver and gold on the International Temperature Scale of 1990. These are independent thermal references, unlike a test that checks only a voltage-input circuit.^[ref]

For a concrete electrical check, the SCC-TC guide describes shorting the input after 30 minutes of powered operation and adjusting its offset to a zero-voltage readout. Its separate calibrator procedure requires equal connection temperatures at the calibrator and module, measures the supplied voltage, converts it to temperature, and compares with the calibrator setting. Such an injected-signal test checks the readout chain; it does not expose a real measuring junction to an independently known bath temperature.^[ref]

Useful additional checks follow from the model: log the raw voltage, the reference temperature, and the final inferred temperature separately; hold $T_h$ stable while changing $T_c$ and test whether cold-junction compensation preserves the inferred $T_h$; compare against an independent thermometer at several stable heat bath settings. Check immersion and repeatability instead of assuming a steady display establishes thermal equilibrium. These are proposed diagnostic checks, not new experimental results. The equal-temperature and polarity controls are already described under Seebeck effect.

Limits arise separately from the thermocouple reference function, sensor-wire changes, thermal contact, terminal temperature differences, and the voltage-input circuit. In particular, the SCC-TC guide warns of connector-to-module temperature gradients and voltage drops caused by the fault-detection electric current flowing through lead electrical resistance.^[ref] A fitted curve or compensation algorithm cannot make all these errors vanish. No single accuracy is assigned to every thermocouple.

The thermocouple reference junction is the connection region where the two thermoelements meet the measurement leads. In ordinary copper-lead wiring there are two material transitions, A–Cu and B–Cu, rather than a second welded A–B junction. They must have a common known temperature $T_c$ for the simple expression $E(T_h,T_c)$ to apply. An ice-water heat bath can establish a nominal $0{}^{\circ }\mathrm{C}$ reference; cold-junction compensation instead measures $T_c$ locally. The word “cold” describes the reference role and imposes no requirement that $T_c<T_h$.

The thermocouple measuring junction joins A and B at the location whose temperature is to be inferred. Its own temperature is what the electrical measurement estimates. To infer a surrounding object's temperature, the junction must approach thermal equilibrium with that object: insufficient immersion, heat conduction along the wires, or a changing environment can keep the junction at another temperature. A stable electrical display alone does not establish that this thermal condition holds. The two-heat bath procedure and terminal-reversal controls under Seebeck effect provide the starting experiment; the practical issue here is the reference and conversion chain.

A thermopile connects multiple thermocouples in series to turn a temperature difference into a larger electrical signal. Absorbed thermal radiation can heat its sensing junctions while reference junctions remain cooler; the signal depends on the thermal and optical design.

A thermopile is an array of thermocouples connected as a series circuit with their electromotive forces adding in the chosen terminal order. It measures a collective temperature difference through the Seebeck effect. An open-circuit electric potential difference requires no externally supplied electric current through the array: a high-input-electrical resistance voltmeter reads the signal. The heat that maintains the junction temperature difference still comes from somewhere, such as absorbed electromagnetic radiation.

In a membrane detector, an absorbing film heats the measuring junctions while reference junctions lie on the surrounding support. Actual junction temperatures can differ across the membrane.^[ref] This converts difficult-to-observe heat transfer into an accessible electrical signal; the series connection combines many small contributions without requiring every thermocouple to have a separate meter.

A capacitance diaphragm gauge senses pressure through deflection of a diaphragm that changes a capacitor's capacitance. Its head can have a different temperature from the vessel being measured, allowing thermal transpiration in the connecting tube to change the head pressure. ^[ref]

Free bending has two zero-energy filament modes, so it does not define an unrestricted normalizable thermal height distribution. Fix the mean displacement and linear tilt, or project out that affine kernel. On the positive complement, the equipartition theorem gives independent Gaussian mode variances $k_{B}T/(A{k}_n^4)$ for normalized modes. The resulting covariance is the bending operator pseudoinverse multiplied by $k_{B}T$. It depends on how the rigid motion is fixed.

With fixed pressure $p$, let $A=\mu p/\mathcal{R}$, so $\rho =A/T$. A material mass coordinate has ${\partial }_x=\rho {\partial }_m$ and the material time derivative becomes ${\partial }_t{|}_m$. The energy and continuity equations then give the displayed equation with $\tau =Ct$ and $C=(\gamma -1)(\mu /\mathcal{R}{)}^{2}p/\gamma$. A mass integral from a fixed origin is material only when the mass flux there vanishes; otherwise subtract its time-integrated flux to label parcels. Uniform pressure is a low-Mach approximation when sound crossing is faster than thermal evolution.

In the absence of heat advection, an insulating sphere of radius $a$ in a uniform remote temperature gradient $\mathbf{H}$ has the displayed exterior harmonic function as its temperature field. The reflected dipole term makes ${\partial }_{r}T=0$ at $r=a$. The surface temperature is $T_0+(3/2)a\mathbf{H}\cdot \mathbf{n}$, so the tangential gradient is amplified by $3/2$. If the sphere has thermal conductivity ratio $K$ relative to its surroundings, temperature and heat-flux continuity instead give surface gradient factor $3/(2+K)$.

Thermal resistance measures the temperature drop required to drive a given rate of heat transfer. For the steady one-dimensional bar model in thermal conduction,Here $r$ relates the temperature drop to heat flux, while $R_{\mathrm{th}}$ relates it to the rate of heat transfer. The single bar above has $r=0.050{\mathrm{m}}^2\mathrm{K}{\mathrm{W}}^{-1}$ and $R_{\mathrm{th}}=5\mathrm{K}{\mathrm{W}}^{-1}$, so its $20\mathrm{K}$ drop drives $4\mathrm{W}$.

To see why series thermal resistances add, apply Fourier's law in each layer to obtain $\Delta T_i=\dot{Q}{R}_i$. With no accumulation or side losses the rate of heat transfer is the same through both, so adding the drops gives $T_h-T_c=\dot{Q}(R_1+R_2)$. This lets us size layered walls without solving a new differential equation for every layer. The structure matches Ohm's law: temperature difference plays the role of electric potential difference, the rate of heat transfer plays the role of electric current, and thermal resistance plays the role of electrical resistance. ^[ref]

A tetrahedron whose six edges have equal length; its four faces are equilateral triangles.

For independent edge vectors $u=B-A$, $v=C-A$, $w=D-A$, the displacement $x=O-A$ solves $x\cdot u=|u{|}^2/2$, $x\cdot v=|v{|}^2/2$, $x\cdot w=|w{|}^2/2$. The reciprocal cross-product basis gives the displayed formula. Its squared distances to all four vertices are equal. It need not lie inside the tetrahedron.

The center of the sphere tangent to all four faces is the positive face-area-weighted combination of the opposite vertices. If $V$ is the volume and $S$ the total face area, its distance from each face is $3V/S$. Indeed its weight for vertex $A$ is $S_A/S$ and the corresponding altitude is $3V/S_A$, so their product is the common distance. Positive weights place it inside the tetrahedron.

Place the centroid of a tetrahedron at the origin, so its four vertex vectors satisfy $r_1+r_2+r_3+r_4=0$. Midpoints of opposite edges have opposite vectors, hence equal distances from the centroid. For the three opposite-edge pairs, choose midpoint vectors $(r_1+r_j)/2$, $j=2,3,4$. Expanding their squared lengths and using ${\sum }_{j=2}^4{r}_j=-r_1$ givesThe argument uses no regularity or equal-edge assumption on the tetrahedron.

For the tetracode length-isospectral lattice construction $L_{\pm }=(\pm 3I+S){\mathbb{Z}}^4$, choose the first coordinate $i$ divisible by three and reverse its sign. Such a coordinate always exists because each nonzero tetracode word has one zero coordinate. The reflection preserves the reduction modulo three and switches the two lattice membership conditions modulo four. It is an involution between $L_{+}$ and $L_{-}$ preserving every absolute coordinate, so arbitrary positive diagonal weights preserve equality of their vector-length multisets.

Add respectively the conditions ${\sum }_i{x}_i\equiv 0(\mathrm{mod}4)$ and ${\sum }_i{x}_i-2x_1\equiv 0(\mathrm{mod}4)$ to the displayed common conditions. These define the two integer Euclidean lattices underlying the tetracode length-isospectral lattice construction. They coincide on their all-even vectors and differ on all-odd vectors. Reflecting a coordinate divisible by three preserves the code condition; on odd vectors it changes the coordinate sum by two modulo four, while on even vectors both conditions remain satisfied. This yields the coordinate reflection bijection for tetracode lattices.

Choose positive axis weights linearly independent over the rationals, and suppose both integer Euclidean lattices contain $q{\mathbb{Z}}^d$ for some positive integer $q$. A length-preserving linear map sends each $q{e}_i$ to an integer vector with exactly that absolute-coordinate pattern, by the displayed implication. Thus the map must be a diagonal sign change. For the four-coordinate tetracode, preserving the code forces all four signs to agree, because each of its four nonzero projective words has support on a different triple of coordinates. A global sign cannot exchange the two odd-vector parity conditions, proving nonisometry for these weights.

Consider a point mass of fixed mass $M>0$ released from rest in a stationary medium. Choose downward as positive, assume constant gravitational acceleration $g>0$, neglect buoyancy, and impose linear drag $F_d=-bv$ with constant linear friction coefficient $b>0$. Here $v$ is the signed downward velocity component; it equals speed during this descent. Newton's second law becomesThe terminal velocity balances the two forces, and the time constant $\tau$ controls how quickly that balance is approached:For the illustrative parameters $M=1\mathrm{kg}$, $b=2kg/s$ and $g=9.8m/{\mathrm{s}}^2$, the time constant is $0.5\mathrm{s}$ and the terminal velocity is $4.9m/s$ downward. At $t=1\mathrm{s}=2\tau$, the downward velocity is $4.9(1-e^{-2})m/s\simeq 4.23686m/s$ and the downward acceleration is $9.8e^{-2}m/{\mathrm{s}}^2\simeq 1.32629m/{\mathrm{s}}^2$. This is a toy linear drag model, not a claim that every $1\mathrm{kg}$ falling object has this resistance law.

For a proof outline, subtract the equilibrium value: $u=v-v_T$ obeys $du/dt=-u/\tau$. The exponential decay solution and the initial condition give $u=-v_T{e}^{-t/\tau }$; one derivative gives the acceleration. Every time constant multiplies the remaining gap $v_T-v$ by $e^{-1}$. The exact limit is reached only as $t\to \infty$. This makes the model useful for estimating a settling duration without mistaking vanishing acceleration for vanishing velocity.

Newton already analyzed vertical motion under uniform gravity and resistance proportional to velocity in Book II, Section I, Proposition III of the Principia. His geometric construction includes a greatest downward velocity, illustrating that a limiting rate can emerge from balancing a driving force and resistance. ^[ref]