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The Lagrange multiplier theorem for one regular constraint says: if $f,k$ are $C^1$ near $x$, $f(x)=0$, $\nabla f(x)\ne 0$, and $x$ is a local extremum of $k$ restricted to $f^{-1}(0)$, thenFor several constraints with linearly independent gradients, the objective gradient lies in their linear span. The regularity assumption rules out singular points where the usual multiplier conclusion can fail.

For an increment $h$, direct expansion and the chain rule giveFor example, $\Vert x+h{\Vert }^2-\Vert x{\Vert }^2=2x^{T}h+\Vert h{\Vert }^2$, so the quadratic remainder divided by $\Vert h\Vert$ tends to zero. The corresponding gradients are $\nabla g(x)=2x$ and $\nabla k(x)=2A^{T}Ax$; the derivative of $F$ is the constant matrix $A$.

The Lagrange multiplier theorem for one regular constraint says: if $f,k$ are $C^1$ near $x$, $f(x)=0$, $\nabla f(x)\ne 0$, and $x$ is a local extremum of $k$ restricted to $f^{-1}(0)$, thenFor several constraints with linearly independent gradients, the objective gradient lies in their linear span. The regularity assumption rules out singular points where the usual multiplier conclusion can fail.

Spontaneous symmetry breaking means that a pure thermodynamic phase is less symmetric than the Hamiltonian even when the selecting external field is removed. For example,The direction of the infinitesimal field is held fixed in this limit. Reversing the limits in a finite symmetric sample normally gives zero. For the XY model, an ordered state chooses one direction from a circle of energetically equivalent directions.

A Goldstone mode is a long-wavelength fluctuation along this continuously degenerate family. A uniform angle rotation costs no energy; a slowly varying one costs only gradient energy. In the stated static phase-only model the quadratic kernel is $\bar{K}{q}^2$, tending to zero as $q\to 0$. There is one independent soft order-parameter phase coordinate for broken $O(2)$ rotation. The static Hamiltonian does not specify a dynamical dispersion $\omega (q)$; that requires a dynamical law. Magnitude fluctuations instead have a restoring local curvature and have been excluded from this low-energy model.

A coarse-grained scalar order parameter $\varphi (\mathbf{x})$ describes a short-range Ising-like transition with sign symmetry. Write the dimensionless Landau-Ginzburg functional asHere $r=a_{t}t$, with the relevant critical-mass shift understood when fluctuations are included. Higher local powers and gradients are negligible only within a justified long-distance expansion. The partition function is $Z=\int \mathcal{D}\varphi e^{-\mathcal{H}}$ with a microscopic cutoff. The local functional is not already the fluctuation-integrated thermodynamic free energy.

Mean-field theory replaces this integral by its stable uniform saddle, $r{\varphi }_0+u{\varphi }_0^3=h$. At $h=0$,The branch curvature is $r_L=r+3u{\varphi }_0^2$: it is $r$ above and $2|r|$ below. HenceTogether with the finite heat-capacity jump from $f_{\min }$, this yieldsThe equal-$|t|$ susceptibility amplitude ratio is two. Uniform saddles also miss domain walls and spatial texture. For an $n$-component order parameter, symmetry allows a vector generalization; ordered transverse directions additionally carry Goldstone fluctuations.

To include small fluctuations set $\varphi ={\varphi }_0+\delta \varphi$. The Gaussian action is $\frac{1}{2}\int [c(\nabla \delta \varphi {)}^2+r_L(\delta \varphi {)}^2]$, givingThe displayed $G(q)$ is a spectral covariance per Fourier delta, avoiding an extra volume factor. As $r_L\to 0$, a growing correlation length lets increasingly large correlated regions fluctuate. The singular Gaussian heat correction contains $\int d^{d}q/(c{q}^2+r_L{)}^2$, which is infrared singular for $d<4$ and logarithmic at four. This signals failure of a simple saddle expansion; the Gaussian heat exponent obtained by extrapolation is not the exact interacting critical exponent.

The Ginzburg criterion compares the fluctuation of a correlation-volume average with ${\varphi }_0^2$, rather than using all ultraviolet fluctuations at a point. Keeping $q\lesssim {\xi }^{-1}$ below the transition gives, up to finite convention-dependent constants,Dividing by ${\varphi }_0^2=|r|/u$ yieldsFor $d<4$, writing $r=a_{t}t$ gives a nonuniversal Ginzburg reduced-temperature windowInside $|t|\lesssim \mathrm{Gi}$, interacting fluctuations must be treated. For example, in three dimensions $\mathrm{Gi}\sim u^2/(c^3{a}_t)$ in this normalization. Above four dimensions the long-wavelength ratio vanishes at criticality; at four it is marginal and interaction-generated logarithms require further analysis.

The spatial gradient term supplies correlations absent from uniform Landau theory. The Ginzburg test bounds where its mean-field saddle remains self-consistent; it does not compute nonclassical exponents or a universal material-independent window.^[ref]

Renormalization compares the same long-distance physics at different observation scales. The starting object is a microscopic or cutoff statistical Hamiltonian ${\mathcal{H}}_g$ with every symmetry-allowed coupling $g$. Because a critical correlation length is much larger than a lattice spacing, repeatedly eliminating short-distance detail can reveal scale-invariant behavior.

In a real-space transformation choose a normalized blocking kernel $B(S|s)$ for retained variables. DefineSumming over $S$ proves exact preservation of $Z$ before truncation. The transformed Hamiltonian generally gains longer-range and many-body interactions and a field-independent constant. Observables must be blocked consistently. In momentum-shell language split $\varphi ={\varphi }_{<}+{\varphi }_{>}$, integrate $\Lambda /b<|q|<\Lambda$, then use $x^{\prime }=x/b$ and a field rescaling to restore cutoff and kinetic normalization. Coarse-graining, rescaling and tracking generated interactions are distinct steps.

Repeated transformations generate a flow $g^{\prime }=R_b(g)$ with composition $R_b{R}_c=R_{bc}$ when the same exact coarse-graining family is used. Losing degrees of freedom generally makes this a semigroup, despite its conventional group name. A renormalization-group fixed point is invariant after rescaling. At a continuous transition the large-distance theory approaches a critical renormalization-group fixed point; finite correlation length instead eventually flows into an ordinary thermodynamic phase regime.

Linearize in scaling coordinates $u_i$:For increasing observation length, $y_i>0$ is relevant and must be tuned, $y_i<0$ is irrelevant, and $y_i=0$ is marginal and needs nonlinear analysis. Discrete multipliers $b^{y_i}$ must be compared with one, not with zero. The stable manifold forms the critical surface after relevant directions have been set to zero.

After subtracting regular additive contributions, ordinary hyperscaling givesChoosing $b=|t{|}^{-1/y_t}$ or choosing the field as scale yieldsFor a critical two-point function $G(r)\sim r^{-(d-2+\eta )}$, the order parameter scaling dimension is $x_m=(d-2+\eta )/2$, and its conjugate field has $y_h=d-x_m=(d+2-\eta )/2$. This recovers the Q1 identities while distinguishing the gap exponent $\Delta$ from the field's scaling dimension $y_h$.

As a concrete dimensional check, start from a scalar gradient action. Restoring its coefficient uses ${\varphi }^{\prime }(x^{\prime })=b^{(d-2)/2}{\varphi }_{<}(b{x}^{\prime })$, so at the Gaussian fixed pointQuartic interactions are relevant below four dimensions. They can produce an interacting Wilson-Fisher fixed point, whose exponents differ from naive Gaussian counting. Above four, the quartic coupling is irrelevant at the Gaussian fixed point but dangerous for ordered observables: ${\varphi }_0^2=-r/u$ is singular when $u$ is dropped. Thus the simple two-variable hyperscaling formula is not universal above the upper critical dimension.

Different microscopic systems can flow to the same long-distance renormalization-group fixed point, explaining universality of exponents and normalized scaling forms. Critical temperatures, metric factors and regular free energies remain nonuniversal. Dimension, order parameter symmetry, interaction range and additional soft variables constrain the universality class. Truncations, perturbative expansions and numerics approximate the flow; they must not be confused with the exact elimination definition.^[ref]

Empirical equivalence must be specified relative to a domain and an account of what counts as observable. Two theories may fit all evidence collected so far while predicting different future results; that is ordinary underdetermination by a finite record. A stronger equivalence requires the same observable consequences for every permitted experiment in the stated domain. For stochastic predictions, matching a set of possible outcomes is not enough: the theories must assign the same outcome probability distributions under corresponding preparations and testing assumptions.

A clear physical example at the level of descriptions occurs in electromagnetism. Write the electric field and magnetic field using a scalar potential $\varphi$ and magnetic vector potential $\mathbf{A}$:For a smooth, single-valued function $\chi$, the gauge transformationleaves both fields unchanged: the curl of a gradient vanishes, and the mixed derivatives cancel in the electric-field expression. With the same charges, boundary conditions and force law, these descriptions give the same classical measurements. Agreement is structural, not an accidental fit to a few readings. However, many physicists regard such descriptions as representing one physical situation, so this alone is not an example of two genuinely incompatible physical theories. The smooth global transformation assumption also matters; equality of local fields is not an unrestricted license to identify every globally different potential configuration.

A second example exhibits possible surplus structure. For particles interacting by Newtonian forces depending only on relative positions, a common Galilean transformationleaves relative positions and accelerations unchanged. Accordingly, adding a privileged but completely uncoupled state of uniform rest does not alter any internal mechanical prediction. Rival choices of that unobservable rest state cannot be distinguished by these experiments. This is a deliberately specified completion of the mechanical model: it does not show that rotation or acceleration is undetectable, or that Newtonian mechanics and special relativity are empirically equivalent. The latter give different high-speed predictions. Whether undetectable absolute rest makes a genuinely different theory or a redundant representation is precisely one philosophical issue raised by the example.

Thus there are concrete cases of empirical redundancy, while the existence of equally well-motivated, fully distinct rivals with identical consequences across all physics is a much stronger claim. Historical theories often agree only in limited regimes or after additional assumptions. One must not convert that restricted agreement into an equivalence theorem for every possible test.

The philosophical relevance does not depend on finding a complete historical pair. A constructed rival can expose the logical limits of evidence, although an artificial extension need not deserve the same credibility as an economical account. If two specified models have equal likelihoods for every admissible data set $D$, thenfor finite positive prior odds. The Bayes factor is one; the data do not discriminate between them. Choices about simplicity or interpretation would be additional considerations. Constructive empiricism treats empirical adequacy as the relevant belief commitment in acceptance, while a scientific realist must explain what further warrants belief in the unobservable structure. The equivalence argument identifies a limit of evidential discrimination; it does not establish that every theoretical claim is equally good or that every unobservable claim is false.^[ref]

Gauss's law states that the outward electric flux through a closed surface is the enclosed electric charge divided by the vacuum permittivity:In electrostatics, the electric field is the negative gradient of the electrostatic potential, $\mathbf{E}=-\nabla V$. Substitution into the differential equation form of Gauss's law gives Poisson's equation ${\nabla }^{2}V=-\rho /{ϵ}_0$. Consequently, in a region containing no charge density,which is Laplace's equation. Charges on the bounding conductors enter through the boundary conditions; they need not vanish for Laplace's equation to hold inside the empty gap.

Apply the negative gradient to the leading far-field electric potential. In spherical coordinates,The electric field therefore has the coordinate-independent formFor the electric dipole along $z$, use $V=\alpha z/(x^2+y^2+z^2{)}^{3/2}$. Direct partial differentiation gives the vector componentsOn the positive axis the electric field points along the electric dipole moment; in the equatorial plane it points oppositely, providing a useful sign check.

Take the electric potential to vanish at infinity and work at $r>0$. For a positive point charge $q$ in vacuum,The field is radial and outward. Since a radial scalar function has gradient $\nabla V=(dV/dr)\hat{\mathbf{r}}$,Thus the two expressions have the required vector relationship. Adding an arbitrary constant to $V$ would leave the electric field unchanged; the chosen reference at infinity fixes that constant. The point-charge singularity itself is excluded from the ordinary derivative calculation.

Take $\mathbf{d}$ from the negative to the positive point charge. Their electric dipole moment is $\mathbf{p}=q\mathbf{d}$. Place them at $\pm \mathbf{d}/2$ and use the superposition principle for the electric potential:For $s\gg d$, the first-order Taylor expansion of the reciprocal distances givesto leading electric dipole order. In spherical coordinates aligned with $\mathbf{p}$, the gradient gives $E_s=-{\partial }_{s}V$, $E_{\theta }=-s^{-1}{\partial }_{\theta }V$ and $E_{\varphi }=0$. ThusEquivalently, the electric dipole field is $[3(\mathbf{p}\cdot \hat{\mathbf{s}})\hat{\mathbf{s}}-\mathbf{p}]/(4\pi {ϵ}_0{s}^3)$. The omitted terms vanish relative to these leading expressions as $d/s\to 0$.

For this space-charge-filled parallel-plate capacitor, place the first plate at $x=0$ and the second at $x=d=x_o$, ignoring fringing field of a capacitor. With vacuum permittivity between the plates and fixed charge density $\rho$, Poisson's equation and the prescribed boundary electric potentials giveIntegrating twice and imposing both values yieldsThe electric field is minus the gradient of the electric potential, henceThus the electric potential is a parabola, concave downward for $\rho >0$ and upward for $\rho <0$. The signed electric field is a straight line. Its magnitude has a V-shaped corner if its zerolies inside the gap; otherwise its magnitude is a single straight segment. An interior zero requires $|V_1-V_2|<|\rho |d^2/(2{ϵ}_0)$. At that point the electric potential has its interior extremum. For $\rho =0$, these reduce to a linear electric potential and a uniform electric field.

Take the grounded electrical conductor to occupy $z\le 0$ and approximate the negligible-radius charged sphere by a point charge $Q$ at ${\mathbf{r}}_{+}=D\hat{\mathbf{z}}$. Put a fictitious image charge $-Q$ at ${\mathbf{r}}_{-}=-D\hat{\mathbf{z}}$. The constructed electric potential is zero on $z=0$, has the correct point charge singularity, and vanishes at infinity. The uniqueness theorem for the electrostatic potential makes it the physical electric potential for $z>0$:Taking its negative gradient gives the requested electric field outside the small sphere:On the plane at cylindrical radius $r$, the tangential components cancel and the normal component isThe electrostatic boundary conditions at a conductor, with outward unit normal $+\hat{\mathbf{z}}$ and zero interior electric field, give the induced surface charge densityFinally integrate over annuli of area $2\pi rdr$:The image charge is a computational device, not a real charge below the plane. For a finite conducting sphere, its induced polarization would change the point-source approximation; the stated negligible radius is essential.

In paramagnetism, a material develops magnetization along a weak applied field, with positive susceptibility; in diamagnetism, it develops an induced moment opposing it, with negative susceptibility. In a simple localized-moment paramagnet, Zeeman alignment competes with thermal disorder. For independent two-state moments $\pm \mu$,when the field is weak and demagnetizing corrections are negligible. Stronger fields saturate the moment; interactions and conduction-electron Pauli paramagnetism can depart from this Curie law.

Ordinary diamagnetism comes from a field-induced change of orbital motion, including in closed-shell matter without permanent atomic moments. Its weak susceptibility is often nearly temperature independent away from a transition, and it contributes a background even when another magnetic response dominates. A nonuniform field tends to attract a paramagnetic specimen toward stronger field and repel a diamagnetic specimen in the weak linear regime. Calibrate the holder contribution and field gradient and measure a temperature/field series to distinguish induced response from retained magnetization. Ferromagnetic hysteresis and superconducting flux expulsion require different collective explanations.

stress describes internal force per area. A cut with unit normal $\hat{\mathbf{n}}$ has traction $\mathbf{t}=\mathbf{\sigma }\hat{\mathbf{n}}$; its normal component is $\hat{\mathbf{n}}\cdot \mathbf{\sigma }\hat{\mathbf{n}}$ and its remainder is shear traction. force balance constrains the stress divergence, while ordinary local angular-momentum balance makes the Cauchy stress tensor symmetric in a medium without body couples.

strain describes deformation rather than force. For small displacement gradients,The antisymmetric displacement-gradient part is a rigid rotation to first order and contributes no infinitesimal strain. Engineering shear strain is ${\gamma }_{xy}=2{ϵ}_{xy}$, so it must not be confused with the tensor component. stress has units of pressure; strain is dimensionless.

An isotropic linear elastic solid obeysTwo independent moduli are required. The shear modulus $G$ and bulk modulus $K=\lambda +2G/3$ must be positive for stability. Young's modulus and Poisson ratio satisfyFor a tensile specimen with free side faces, ${\sigma }_{xx}=F/A$ gives ${ϵ}_{xx}={\sigma }_{xx}/E$ and ${ϵ}_{yy}={ϵ}_{zz}=-\nu {\sigma }_{xx}/E$. Pure shear gives $\tau =G\gamma$; hydrostatic loading gives $\Delta V/V=-p/K$. The incompressible limit has $\nu \to 1/2$ at finite shear stiffness, not vanishing shear resistance. Crystals can be anisotropic and need a fourth-rank elastic stiffness tensor instead of the isotropic two-constant law.

elastic deformation is recoverable on unloading; it need not remain linear at large strain. Beyond yield, irreversible rearrangements such as dislocation motion cause plastic strain, often followed by hardening. Ductile samples can neck before fracture; brittle failure has much less plastic deformation. Engineering stress uses initial area, while true stress uses the changing instantaneous area. A decreasing engineering stress during necking therefore need not mean that local true stress decreases. stress concentrations, flaws and loading geometry can determine failure before a uniform-material strength is reached. Viscoelastic materials also show rate dependence, creep and stress relaxation; their losses cannot be described by a static Hooke law alone.

A tensile test should measure force and gauge-length extension independently, correct machine compliance, align the grips, and control strain rate and temperature. Unloading distinguishes recoverable strain from a residual plastic extension. Separate shear or hydrostatic tests determine moduli that one axial measurement does not fix. Compression specimens need friction and buckling checks. Work per volume is $\int {\sigma }_{ij}d{ϵ}_{ij}$; the recoverable part is elastic energy, while a dissipative cycle produces a loss rather than creating energy. The original loading/unloading curve below is a chosen constitutive sketch, not measured material data.

Stokes theorem converts circulation around an oriented boundary into curl flux through a spanning surface. For a sufficiently smooth vector field on an oriented piecewise smooth surface $S$,The boundary orientation follows the right-hand rule relative to $\mathbf{n}$. The field must be smooth on the surface, not merely on its boundary; a singularity crossing the surface cannot be ignored. For a fixed loop, applying the theorem to Faraday's differential law gives the integral flux law of induction.

The planar form is Green theorem. For a region $D$ with positively oriented boundary and sufficiently smooth $P,Q$,The outer boundary is counterclockwise and any holes have clockwise boundaries. For instance $P=-y/2,Q=x/2$ makes the area of $D$ equal to half the line integral of $xdy-ydx$; the unit circle gives area $\pi$.

A closely related use of “Green's theorems” in field theory is Green's identities, obtained by applying the divergence theorem to $\varphi \nabla \psi$:and, after subtracting the expression with $\varphi ,\psi$ exchanged,These relate interior sources to boundary data in electrostatics and other boundary-value problems. For example, a harmonic potential difference vanishing on a Dirichlet boundary condition has zero integral of its squared gradient by the first identity, establishing uniqueness. Both the planar circulation theorem and the field identities are included here to make the convention clear.

A tidal force measures the difference of gravitational acceleration across an extended object. A freely falling reference frame removes a spatially uniform acceleration, but not its gradient. For a distant point mass $M$ at distance $R$ and direction $\hat{\mathbf{n}}$ from the reference centre, a small internal displacement $\mathbf{x}$ givesThe tidal tensor has eigenvalues $2GM/R^3$ radially and $-GM/R^3$ in each transverse direction: radial stretching and transverse squeezing. Its trace vanishes in the source-free region, consistent with the gravitational Gauss law. An associated relative potential is ${\Phi }_{\mathrm{tide}}=-GM[3(\hat{\mathbf{n}}\cdot \mathbf{x}{)}^2-x^2]/(2R^3)$.

The scale is $GMl/R^3$ across size $l$, not the centre's absolute gravity $GM/R^2$. This distinction explains how a more distant massive object can exert a large force but a modest tide. Opposite-side stretching supports the elementary two-bulge picture; real ocean tides also involve rotation, basin geometry and fluid dynamics, so the static gradient alone does not predict coastal heights or timing. The same gradient stresses satellites and can disrupt a self-gravitating body when it competes with the body's own gravitational restoring acceleration.

A fluid is described on a continuum scale by density, pressure and velocity fields. An Eulerian field is measured at fixed position; a material element changes at rate $D/Dt={\partial }_t+\mathbf{v}\cdot \nabla$. mass conservation requiresFor incompressible flow this becomes $\nabla \cdot \mathbf{v}=0$. A streamline is instantaneously tangent to velocity, whereas a pathline follows a particle through time; they coincide in a steady flow but need not in an unsteady one.

For an ideal inviscid fluid, pressure and body forces give the Euler equation. In the constant-density gravitational case,A stationary impermeable boundary requires zero normal velocity, but inviscid theory does not impose no slip. Taking its steady projection along a streamline gives $p/\rho +v^2/2+gz=\mathrm{constant}$. If the flow is also irrotational in the connected region, that constant is shared between streamlines. Inviscid does not mean irrotational: rotating ideal-fluid flows are possible. A velocity potential exists locally for an irrotational field, and in incompressible flow it obeys Laplace's equation. The Bernoulli assumptions must be checked before converting a pressure difference into a speed.^[ref]

Real fluids transmit shear stress. An incompressible Newtonian fluid with constant dynamic viscosity $\eta$ obeysThe constitutive shear stress is proportional to strain rate, and stationary solid boundaries usually impose no slip. viscosity diffuses vorticity and converts organized mechanical energy into heat; its local dissipation $2\eta e_{ij}{e}_{ij}$ is nonnegative. It cannot generally be included merely by changing the constant in a lossless Bernoulli equation.

In a long straight circular tube, steady fully developed laminar flow hasThis Hagen-Poiseuille equation follows by balancing the axial pressure gradient with viscous shear and setting $v_z(a)=0$. The parabolic profile, the fourth-power radius sensitivity and the pumping power $\Delta pQ$ are experimentally distinct consequences. Entrance regions, rough walls, non-Newtonian response or a changing temperature require additional modelling.

The Reynolds number $\mathrm{Re}=\rho UL/\eta$ compares inertia with viscosity. Small values favor creeping-flow balances; high values permit thin boundary layers, separation and turbulence, although transition depends on geometry and perturbations, not one universal critical number. At high Reynolds number an outer flow can be approximately inviscid while near-wall viscosity still determines drag. This resolves why a zero-drag ideal potential-flow model cannot predict the resistance of an ordinary moving body. In turbulent flow, fluctuating Reynolds stresses redistribute momentum and increase the required pressure drop; averaging the equations introduces closure requirements.

A controlled tube experiment measures pressure difference and collected volume per time while changing forcing. Use tracers or a velocity probe to test profile shape, allow sufficient entrance length, and monitor radius and temperature independently. Repeat with reversed flow and at several rates to test offsets and linearity. Dye mixing alone does not measure viscosity or establish a fully developed velocity profile. The original figure below separates a specified laminar velocity profile from an independent pressure/discharge measurement.

Use $x=\rho \cos \varphi$, $y=\rho \sin \varphi$ to recognize the planar quadrupole potential, $V=A(x^2-y^2)$. Its electric field is minus the gradient:or, in cylindrical coordinates,For $A>0$, arrows point toward the origin along the $x$ axis and away from it along the $y$ axis. The off-axis electric field lines satisfyThe axes themselves are the separating field lines; the origin is a field null. Reversing $A$ reverses every arrow. The field is independent of $z$, and $A$ has units $\mathrm{V}{\mathrm{m}}^{-2}$.

Because the magnetic field obeys $\nabla \cdot \mathbf{B}=0$, it can be represented locally as the curl of a magnetic vector potential:This automatically enforces the absence of magnetic monopoles. It is also a global representation throughout ordinary all-space problems without monopole singularities. It is not unique: the gauge transformationleaves $\mathbf{B}$ unchanged because the curl of a gradient vanishes. In a time-dependent problem,makes the same transformation leave both physical fields unchanged and automatically satisfies Faraday's law.

In vacuum magnetostatics, choose the Coulomb gauge $\nabla \cdot \mathbf{A}=0$. Substitution into Ampère's law and the identity $\nabla \times (\nabla \times \mathbf{A})=\nabla (\nabla \cdot \mathbf{A})-{\nabla }^2\mathbf{A}$ givesfor a localized steady electric current density and a potential vanishing at infinity. Taking the curl recovers the Biot-Savart law. The magnetic vector potential converts a constrained vector field problem into componentwise Poisson equations once a gauge is chosen.

For a uniform magnetic field, $\mathbf{A}=\frac{1}{2}\mathbf{B}\times \mathbf{r}$ is a convenient choice. For a localized magnetic dipole moment $\mathbf{m}$, the leading far-field potential isAnother useful relation is Stokes theorem, which relates a closed-path integral to the magnetic flux through a spanning surface:The units of $\mathbf{A}$ are $\mathrm{T}\mathrm{m}$, equivalently $\mathrm{Wb}{\mathrm{m}}^{-1}$.