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Differentiate the Grahame equation for a planar symmetric monovalent electrolyte with respect to surface potential at fixed reservoir concentration and temperature. The result is the displayed capacitance per unit area. It approaches $\epsilon {\kappa }_D$ in the Debye–Hückel approximation. A compact surface layer can contribute a separate series capacitance; point-ion packing assumptions also fail at sufficiently high near-surface concentration.^[ref]

Diffusion is net transport associated with random microscopic motion. A localized patch of a dilute solute can spread through a stationary solvent: individual molecules move in both directions, but more cross from the crowded side of an imaginary surface than from the less crowded side. This imbalance gives a net diffusive flux down the concentration gradient; the molecules do not need to sense which direction has lower concentration. At uniform concentration, microscopic exchanges continue even though the net diffusive flux vanishes. ^[ref]

The distinction from advection is easiest to see in a dye patch. Uniform bulk motion carries its centre downstream; diffusion broadens it relative to that centre. Both can happen together. A moving or spreading patch by itself does not identify the mechanism: a nonuniform velocity field can also stretch a patch. ^[ref]

For a dilute nonreacting solute in a uniform medium at fixed temperature, Fick's first law models the relative diffusive flux as $\mathbf{j}=-D\nabla C$, with scalar diffusion coefficient $D>0$ of units ${\mathrm{m}}^2{\mathrm{s}}^{-1}$. Here $C$ is the amount of substance per unit volume, so $\mathbf{j}$ has units $\mathrm{mol}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$. With carrier velocity field $\mathbf{u}$, the total flux is $C\mathbf{u}+\mathbf{j}$. The continuity equation, with no sources or chemical reactions, givesFor constant $D$ and incompressible flow, this reduces to $C_t+\mathbf{u}\cdot \nabla C=D{\nabla }^{2}C$, where ${\nabla }^2$ is the Laplacian. In a stationary carrier it is the diffusion equation. These assumptions matter: variable $D$ must remain inside the divergence, and changing sources require extra terms. ^[ref]

The useful surprise is the common mathematics: thermal conduction smooths temperature through the analogous heat equation, while molecular diffusion smooths composition. This analogy already motivated the original formulation of Fick's laws. ^[ref] However, the thermal diffusivity and the solutal diffusivity need not be equal. A diffusive time scale $t_D\sim L^2/D$ means that crossing ten times the distance requires roughly one hundred times the time. This is why molecular diffusion is especially effective across thin layers, and why chaotic advection can assist mixing by making those layers thinner. Physical diffusion is a transport mechanism, distinct from the broader probabilistic class of Markov diffusion.

For a linear imposed temperature $T=T_E+Gz$, local liquidus equilibrium gives $C=-(T_E+Gz)/m$ in the mushy layer. With equal densities, pure solid, pulling velocity $-V$, and diffusion only in liquid volume fraction $q=1-\varphi$, steady solute flux is $-VqC-Dq{C}_z=-V{C}_0$. The exterior concentration is $C_0+(C_i-C_0)e^{-V(z-h_0)/D}$. Marginal equilibrium at a mush-liquid boundary gives $C_i-C_0=GD/(mV)$ and hence the displayed thickness. In the mush, $q[C+(D/V)C_z]=C_0$, so $\varphi =(h_0-z)/(m{C}_0/G+h_0-z)$. A positive thickness requires the thermal interval to exceed the diffusive length $D/V$.

For a three-dimensional repulsive gas at zero temperature, leading quantum depletion is $(n-n_0)/n=8\sqrt{n{a}_s^3}/(3\sqrt{\pi })$. Small gas parameter controls the expansion; this expression does not apply unchanged in lower dimensions.

In a dilute gas, Bose-Einstein condensation makes a single one-particle state hold a substantial fraction of the bosons. This is a population effect: cooling reduces the number that the excited energy eigenstates can accommodate at thermal equilibrium. The surplus accumulates in a Bose-Einstein condensate. For an ideal gas the occupied mode is the trap ground state; interactions can deform that mode, as described by the Gross–Pitaevskii equation. Dilute does not mean classical: a small number density can still give overlapping thermal de Broglie wavelengths at sufficiently low temperature.

The appeal is that changing the trap changes a calculable Quantum Hamiltonian, while absorption imaging makes its population distribution visible. The Bose condensation in a three-dimensional harmonic trap below turns density of states into a prediction for a critical temperature and condensate fraction. These are separate questions from measuring matter-wave coherence or superfluidity.

Eric A. Cornell and Carl E. Wieman's joint lecture emphasizes the competition between useful elastic collisions, which restore thermal equilibrium during evaporative cooling, and collisions that remove or heat the trapped sample. A long trap lifetime alone is insufficient: reequilibration must keep up with evaporation.^[ref]

In a thin rectangular electromagnetic cavity, the window $\hslash c\pi /L_x,\hslash c\pi /L_y\ll k_{B}T\ll \hslash c\pi /L_z$ freezes the nonzero thickness indices while retaining many in-plane normal modes. The zero-thickness-index family has one polarization of an electromagnetic wave and density of states $dN=Akdk/(2\pi )=2\pi A\nu d\nu /c^2$. Multiplying by $h\nu /(e^{h\nu /k_{B}T}-1)$ and integrating gives $U_{2D}=4\pi \zeta (3)A(k_{B}T{)}^3/(h^2{c}^2)$. Thus thermal energy scales as area times $T^3$. The Riemann zeta function appears through ${\int }_0^{\infty }{x}^2/(e^x-1)dx=2\zeta (3)$.

This is an intermediate low-temperature regime of an anisotropic finite cavity. When $k_{B}T$ also falls below the first in-plane excitation, thermal energy becomes exponentially small. If a different model excludes every zero thickness index, its first transverse mode has a finite gap and the proposed $T^3$ regime does not occur. In the three-dimensional continuum regime the corresponding density of states is proportional to volume times ${\nu }^2$ and thermal energy is proportional to volume times $T^4$.

The New Jersey Institute of Technology laboratory uses a silicon 1N4148 diode. For its forward current-voltage characteristic, the supply operates in current-source mode; a benchtop voltmeter independently reads the diode electric potential difference. Small electric current steps resolve the steep branch. For reverse measurements, a sensitive ammeter sits in series; the handout suggests a $10$–$100\mathrm{k\Omega }$ series resistor. Reversing the applied polarity supplies the comparison. Meter placement must avoid counting the voltmeter's own electric current, and the zero reading may have an offset. The same diode is then measured with an Agilent U2722A source measure unit for graphical comparison.^[ref]

The prescribed analysis uses a logarithmic scale for forward electric current to infer $I_S$ and $n$ from the straight region. Large-electric current departures can indicate series electrical resistance. The handout cautions that the accessible range may not determine it precisely; meter loading also matters. This is a measurement procedure, without a published set of device readings.^[ref]

The reason for the logarithmic scale follows directly from the Shockley diode equation. If $I_D\gg I_S$ and series electrical resistance is negligible,where $I_0>0$ is a fixed reference electric current making the logarithm dimensionless. The slope is $1/(n{V}_T)$, so extracting $n$ needs the junction temperature. Fitting only an appropriate interval matters: a straight-looking subset cannot establish the model at all electric currents.

For a long, nondegenerate, ideal p-n junction under low injection, the diode saturation current is set by minority carriers supplied from the neutral regions. Their equilibrium densities are $n_{p0}=n_i^2/N_A$ and $p_{n0}=n_i^2/N_D$. Solving the steady diffusion and recombination equations gives $I_S=e\mathcal{A}(D_n{n}_{p0}/L_n+D_p{p}_{n0}/L_p)$, where $\mathcal{A}$ is junction area, $D_n,D_p$ are diffusion coefficients, and $L_n,L_p$ are diffusion lengths. Reverse bias suppresses opposite majority-carrier injection exponentially while collecting this minority supply. Thus $I_S$ is approximately bias independent before breakdown, but depends strongly on temperature through $n_i^2$ and on transport, doping, and geometry.

For a vertical optical transition between parabolic energy bands with common carrier temperature, net stimulated emission has occupation factor $f_c(E_c)-f_v(E_v)$. The monotonic Fermi-Dirac distribution makes this positive exactly when the quasi-Fermi-level splitting exceeds the transition photon energy. An allowed transition must also lie above the band gap. Broadening, band degeneracy and a zero-temperature occupation window require their own qualifications; this condition alone does not establish laser threshold.

The directional derivative tests a function along a chosen straight line, including lines that follow neither coordinate axis. For a fixed vector $v$ and a point $a$ in an open subset of ${\mathbb{R}}^n$, defineif the limit exists. With a unit vector $v$, the parameter $t$ measures signed distance in the input Euclidean space, so the directional derivative is change per unit distance. For a general $v$, it is change per unit parameter; the norm of $v$ also sets the rate of travel.^[ref]

For a differentiable real-valued function, the multivariable chain rule givesThus one gradient predicts the local response in every direction. This formula requires differentiability, rather than merely separate partial derivatives.^[ref]

Return to $f(x,y)=x^2+xy+2y^2$ at $a=(1,1)$. Its gradient is $(3,5)$. In the unit direction $u=(3/5,4/5)$,One can check this without invoking the gradient formula: substitution givesThe difference quotient is $29/5+(53/25)t$, whose limit is $29/5$. This calculation separates the exact change along a finite path from its first-order rate at the starting point.

Changing the travel speed changes the parameter rate. Taking $v=(3,4)=5u$ gives $D_{v}f(a)=29$, because the input travels five distance units per parameter unit. For a differentiable function, $D_{cv}f(a)=c{D}_{v}f(a)$ for any real $c$: negative $c$ reverses the orientation as well as scaling the rate. Normalizing $v$ before comparing different slopes avoids confusing direction with speed.

The Cauchy-Schwarz inequality bounds the rate in any unit direction by $\Vert \nabla f(a)\Vert$. Here the largest rate is $\sqrt{34}\approx 5.831$, attained in the direction $(3,5)/\sqrt{34}$; the chosen $(3/5,4/5)$ gives the slightly smaller rate $5.8$. The perpendicular unit direction $(-5,3)/\sqrt{34}$ gives zero. That means no first-order height change, rather than no change at all: along this direction the example has $f(a+tu)=4+(28/34)t^2$.

This is useful for following a height or temperature scalar field, choosing a local search direction in mathematical optimization, and distinguishing a coordinate sensitivity from a response along an actual path. For a curved path, derivative along a curve uses its instantaneous tangent vector. Even existence of all directional derivatives does not guarantee one common linear approximation; directional derivatives do not imply differentiability gives a concrete counterexample.

Directional emissivity is the ratio of emitted radiance to blackbody radiance at the same temperature in a specified outgoing direction. It can be total over wavelength or additionally spectrally resolved. Normal-direction emissivity is not automatically total hemispherical emissivity: the latter integrates all directions with the projected area weighting. A diffuse thermal emitter removes this directional dependence of emitted radiance.

A direction field for $y^{\prime }=f(t,y)$ places a short segment with slope $f(t,y)$ at each sampled point where $f$ is defined. A function graph of a solution of a differential equation follows these segments: its tangent line has exactly the prescribed slope. Segment length is a drawing choice, rather than a speed; arrows, if used, point toward increasing $t$. An isocline collects points with one common slope.

For Newton's law of cooling, $y^{\prime }=-0.1(y-20)$ with time in minutes and temperature in degrees Celsius, the slopes at $y=80,40,20,10$ are respectively $-6,-2,0,1$ degrees Celsius per minute. Each horizontal row is an isocline, because this autonomous differential equation has no explicit time dependence. Above $20$ the solutions of a differential equation descend, below $20$ they rise, and $y=20$ is an equilibrium point.

The highlighted initial condition $y(0)=80$ selects one curve from the family. The picture reveals approach to the equilibrium point before any algebra is done. Under the hypotheses of the Picard-Lindelöf theorem, distinct function graphs of solutions cannot cross or even meet in the time-state plane. A direction field alone does not guarantee uniqueness: several curves can share a tangent line at the same point when those hypotheses fail. See the delayed-start example under initial value problem.

This geometric interpretation is developed in Direction Fields, Isoclines, Integral Curves.

Suppose $k>0$ and two sufficiently regular steady state temperature profiles solve the same heat equation with the same PDE source term $H$ and the same Dirichlet boundary condition $h$. Their difference $w$ satisfies $\nabla \cdot (k\nabla w)=0$ and $w=0$ on the boundary. The divergence theorem applied to $wk\nabla w$ givesThus $\nabla w=0$, and the zero boundary values force $w=0$ on each connected component. The argument proves uniqueness without asserting existence and without replacing the spatially varying thermal conductivity by a constant. For weak solutions, the same calculation is the energy argument obtained by choosing $w$ as the test function, under the usual boundedness and positive lower bound assumptions on $k$.

James F. Price's Draining a Tank, an appendix to Dimensional Analysis of Models and Data Sets, describes an actual tabletop experiment in sections 2.1–2.3. Tap water filled a semitransparent polypropylene beaker with a marked volume scale and drilled side openings. Some trials added horizontal outlet tubes. Temperature was measured, and dye checked that the water had settled before uncovering the outlet. The readout was elapsed time at successive volume marks. Opening diameter and tube length varied between trials. Reported repeatability was checked; tube bores were measured with feeler gauges rather than accepted from nominal sizes.^[ref]

For tank area $A$ and volume $W_o$ at outlet level, the observations give $h_i=(W_i-W_o)/A$ and interval volumetric flow rate $Q_i\simeq (W_i-W_{i+1})/(t_{i+1}-t_i)$. Price compared $Q/(a\sqrt{2gh})$ across heads to infer a discharge coefficient where that ratio was nearly constant. At small head, drainage could stop with a protruding water cap; surface tension mattered. Outlet tubes introduced viscosity effects. This experiment tests a useful operating range, rather than proving ideal drainage down to zero head.^[ref]

A useful independent test of the constant-coefficient model avoids numerical derivatives of noisy head readings. Its exact square root relation predictsFit a straight line to $\sqrt{h_i}$ against time over a chosen positive-head interval; the negative slope determines $b$. Repeat from the same initial head and compare slopes, then vary the head interval to test constancy of $C_d$. This is a proposed diagnostic derived from the draining tank with constant cross-sectional area, not an additional procedure or measured fit attributed to Price. Startup transients, uncertain outlet level, a changing jet shape and terminal low-head data can spoil the straight line. Independent collection of discharged volume provides a check on the tank-scale readout.

A disk flares when its disk aspect ratio increases outward. This is stronger than the statement that its height increases: a disk with $H\propto r$ has constant aspect ratio and is not flaring by this definition. The radial temperature and surface density of a disk profiles determine flaring in a specified vertical equilibrium model.

Suppose molecular hydrogen is fully dissociated into atoms across a steady planar shock, with the same far-upstream and far-downstream temperature and negligible binding-energy mass correction. The upstream isothermal sound speed obeys $a_1^2=k_{B}T/(2m_H)$, while $a_2^2=2a_1^2$. Conservation of mass gives $\chi ={\rho }_2/{\rho }_1=u_1/u_2$. Momentum conservation then gives $u_1^2+a_1^2=u_1^2/\chi +2a_1^2\chi$, producing the displayed quadratic with ${\mathcal{M}}_1=u_1/a_1$.

The high root approaches ${\mathcal{M}}_1^2/2$ at large upstream Mach number. Its real compressive domain is ${\mathcal{M}}_1\ge 1+\sqrt{2}$. At the endpoint the downstream flow is sonic; above it the high root is subsonic downstream. Completion of dissociation and maintenance of equal temperatures additionally require a consistent chemical and radiative energy budget. The two flux equations alone do not establish that physical realizability.