A particle constrained to a surface can have a stable minimum of electrostatic potential restricted to that surface even when the unrestricted equilibrium is unstable. Away from source charges, Laplace's equation makes the full potential Hessian matrix trace zero, which rules out a positive-definite three-dimensional Hessian but does not rule out a positive-definite restriction to two dimensions. Equal positive charges at the corners of a square give a planar minimum at its centre and an unstable perpendicular direction.

An electrical reference potential fixes the arbitrary additive constant of an electric potential. Assigning zero to ground makes other potentials differences from that baseline; it does not assert that the electric field vanishes there. If a source calculation gives $\Phi (a)=89.9\mathrm{V}$ and $\Phi (b)=179.8\mathrm{V}$ relative to infinity, choosing $a$ as the zero instead gives ${\Phi }^{\prime }(a)=0$ and ${\Phi }^{\prime }(b)=89.9\mathrm{V}$, with the same electric potential difference. Grounding an electrical conductor additionally connects it to a reservoir that can exchange electric charge to maintain the prescribed electric potential; that physical boundary condition differs from merely relabeling an unchanged solution.^[ref]; ^[ref].

Conductive-paper electric potential mapping uses a resistive sheet as a two-dimensional analog of an electrostatic boundary-value problem. Painted or taped conducting electrodes impose selected electric potentials. One voltmeter contact stays at a reference electrode while a movable contact locates points at the same electric potential; joining those points gives an equipotential map.

The analogy follows from steady electric charge conservation and local Ohm's law. In a source-free sheet of uniform isotropic electrical conductivity $\sigma$,The measured potential therefore satisfies Laplace's equation. In-plane electric field components follow from $\mathbf{E}=-\nabla V$; more closely spaced equal-potential contours indicate a larger local field magnitude. Nonuniform conductivity, electrode contacts and voltmeter loading can alter the inferred solution. The apparatus models a two-dimensional geometry, not every finite three-dimensional electrode arrangement. ^[ref]

A voltmeter measures an electric potential difference between two contacts; its finite input electrical resistance can cause voltmeter loading.

The trace-free electric quadrupole moment is a symmetric tensor measuring the second-order term in an electric multipole expansion. For a localized charge distribution its contribution to the far-field electrostatic potential is $Q_{ij}{\hat{r}}_i{\hat{r}}_j/(8\pi {ϵ}_0{r}^3)$, with summation over indices. The moment may be nonzero even when the total electric charge or electric dipole moment is nonzero; it should not be confused with an electric quadrupole whose lower moments vanish. Changing the origin generally changes the tensor when lower moments are nonzero.

For an ohmic element, electrical resistance is $R=V/I$ and conductance is its reciprocal $G=I/V$. They describe the same voltage-current response using reciprocal units. The relation $V=IR$ is Ohm's law.

The ampere is the SI unit of electric current. Since 20 May 2019 its definition fixes the elementary charge exactly at $e=1.602176634\times 10^{-19}\mathrm{C}$, with $\mathrm{C}=\mathrm{A}\mathrm{s}$. This gives electrical measurements a reference based on a fixed elementary charge. ^[ref]; ^[ref].

Conventional current uses the direction in which positive electric charge flows. A moving negative electric charge contributes electric current in the opposite direction to its velocity. Thus electrons drifting left produce conventional current to the right; the convention describes signed electric charge transport, without requiring positive charge carriers. ^[ref].

A galvanometer detects electric current through a mechanical response, such as the deflection of a pointer.^[ref]

A sensitive galvanometer makes a small electric current visible; a calibrated deflection also estimates its magnitude. A center-zero scale distinguishes both signs of electric current. The moving-coil galvanometer rotates a coil in an imposed magnetic field, whereas a moving-magnet galvanometer rotates a magnetic needle in the coil's magnetic field. The spring-balanced model below describes the moving-coil galvanometer, not every instrument called a galvanometer.

An ammeter measures electric current through a branch of an electrical circuit. Connect it in series; its own finite electrical resistance changes the measured branch's electric current unless that effect is negligible or corrected.^[ref]

For real frequency $\omega >0$ and both vertical wavenumbers $p=k_{\alpha }>0$, $s=k_{\beta }>0$, let $D=k^2+ps$, $A=\rho \omega p/D$, $B=\rho \omega s/D$, and $C=k(2\mu D-\rho {\omega }^2)/(\omega D)$. With traction $\mathbf{t}=({\sigma }_{xz},{\sigma }_{zz})$ and particle velocity $\mathbf{v}=(v_x,v_z)$, each directional P wave/SV-wave field obeys $\mathbf{t}=-Z_{\pm }\mathbf{v}$. Direct displacement-potential substitution yields $Z_{-}=-Z_{+}^{†}$. The Hermitian part of $Z_{+}$ is positive diagonal; thus forward/backward fluxes have opposite signs and mixed directional flux cross terms cancel.

The intrinsic signed curvature is the curvature of a stress-free filament, measured along material arc length. An elastic filament with actual curvature $\kappa$ stores bending energy $\frac{B}{2}\int (\kappa -{\kappa }_0{)}^{2}ds$, with filament bending modulus $B$. In a planar small-slope Monge representation, $\kappa \simeq {\zeta }^{\prime \prime }$ and the leading energy is quadratic in ${\zeta }^{\prime \prime }-{\kappa }_0$.

For an isotropic elastic rod with shear modulus $G_s$, polar second moment of area $J_p$, and twist rate $\Omega$, the torque is $C\Omega$ and the energy per length is $C{\Omega }^2/2$. For a circular section of radius $r$, $J_p=\pi r^4/2$. This material twist stiffness is distinct from the energy of bending filament centerlines into a helix.

Let a convex strain energy density have derivative equal to the actual stress tensor. Under mixed displacement and traction data, any equilibrated stress ${\sigma }_0$ with the prescribed body force and boundary tractions represents the external work. For any compatible trial strain ${\epsilon }^{\prime }$, the convex supporting-plane inequality gives $W({\epsilon }^{\prime })-W(\epsilon )\ge \sigma :({\epsilon }^{\prime }-\epsilon )$. The integral of $(\sigma -{\sigma }_0):({\epsilon }^{\prime }-\epsilon )$ vanishes by integration by parts. Thus the actual strain minimizes $\int (W-{\sigma }_0:\epsilon )$.

The Einstein angular radius is the angular form of the Einstein radius: $R_E=D_l{\theta }_E$ in the lens plane. Its displayed value applies to a point-mass gravitational lens with the angular diameter distances defined above. It depends on source distance as well as lens mass and position. At fixed distances, multiplying $M$ by four doubles ${\theta }_E$.

At exact alignment, the vector thin gravitational lens equation is $0=(1-{\theta }_E^2/r^2)\mathbf{\theta }$. For a nonzero image direction it requires $r={\theta }_E$ and leaves the direction unrestricted: all azimuths form an Einstein ring. The signed equation's roots $\theta =\pm {\theta }_E$ are only a diameter cut through that circle. A source of finite angular size makes an annulus or arcs with finite received light; the infinite point-source lensing magnification at alignment is an idealization. An Einstein ring expresses rotational symmetry and source alignment, not a circular arrangement of emitting stars.

For a nondegenerate orthonormal coframe in spacetime $E^a$ and independent antisymmetric Lorentz connection 1-form, the action $\int {ϵ}_{abc}(R^{ab}\wedge E^c+\lambda E^a\wedge E^b\wedge E^c)$ has independent coframe and connection variations. Since $\delta R^{ab}=D\delta {\omega }^{ab}$, integration by parts yields a connection equation ${ϵ}_{abc}{T}^c=0$. The three variations of the cubic coframe term give the coframe equation ${ϵ}_{abc}(R^{ab}+3\lambda E^a\wedge E^b)=0$. In three dimensions these are exactly the equations displayed above. The solution has constant sectional curvature $K=-3\lambda$. Omitting the factor of three changes the cosmological normalization.

An independent vierbein and metric-compatible spin connection allow spin density to source a torsion tensor. The connection equation from the Einstein-Hilbert action is algebraic in torsion. In the absence of connection-dependent matter it fixes the Levi-Civita connection. For a minimally coupled Dirac field, eliminating the connection generates a local four-fermion interaction. Thus its equivalent torsion-free second-order action includes this contact term; simply dropping it defines a different matter theory.

Local Lorentz invariance requires the outcome of a local experiment, expressed in its freely falling apparatus, to be independent of the apparatus's velocity and orientation. In the Einstein equivalence principle it applies to nongravitational experiments; the strong equivalence principle also includes local gravitational experiments. It is distinct from local position invariance.

Local position invariance requires identical local experiments to have the same outcome independently of where and when they are performed. For nongravitational experiments it is part of the Einstein equivalence principle. Comparisons of clocks through gravitational redshift test this principle once the gravitational potential difference is independently specified; they do not by themselves test the whole strong equivalence principle.

In the ideal-gas high-temperature limit, the Standard Model has 28 boson states: 16 gluon states, eight electroweak gauge boson states and four Higgs field states. Its 90 fermion states are 72 quark states, 12 charged lepton states and six neutrino states. Counting a relativistic fermion state with weight $7/8$ gives the displayed effective number of relativistic entropy degrees of freedom. The effective number of relativistic energy degrees of freedom has the same value when all species share a temperature. Mass thresholds and interactions change the ideal-gas count.