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An ideal Gaussian chain is a polymer model whose independent segment vectors have zero mean and isotropic Gaussian distribution. If $⟨|\mathbf{r}{|}^2⟩=b^2$, each Cartesian variance is $b^2/3$, so a displacement across $\ell$ links has density $(3/(2\pi \ell b^2){)}^{3/2}\exp [-3R^2/(2\ell b^2)]$ and characteristic function $\exp [-k^2{b}^2\ell /6]$. The continuous-contour version uses the same characteristic function for any contour separation. It is the large-scale limit of a freely jointed chain, while allowing unbounded Gaussian segment lengths and omitting interactions between distant parts of the chain.

Rolling a sheet into a circular cylinder bends it without changing its internal lengths and angles. Fitting that same sheet smoothly onto a sphere requires distortion of its first fundamental form. Gaussian curvature distinguishes these situations: it measures the intrinsic curvature of a smooth surface, even though its simplest definition uses how the surface bends in Euclidean space.

The Gaussian curvature at a point is the determinant of the shape operator, or the product of the two principal curvatures:Here $I=Ed{u}^2+2Fdudv+Gd{v}^2$ is the first fundamental form and $II=ed{u}^2+2fdudv+gd{v}^2$ is the second fundamental form, with $S=-dN$. Regularity makes $EG-F^2>0$. Reversing the unit normal negates $e,f,g$ and both principal curvatures, leaving Gaussian curvature unchanged. ^[ref]

The sign describes bending at the point: positive Gaussian curvature means the principal curvatures have the same nonzero sign, as at an elliptic point; negative Gaussian curvature means opposite signs, as on a saddle; zero Gaussian curvature means at least one principal curvature vanishes. Zero Gaussian curvature at a single point does not imply that a neighborhood is a plane. ^[ref]

Four worked models make the signs concrete:

The patches use a common length unit, with $R=1$ for the sphere and circular cylinder. The marked red and blue curves have principal directions at the black point; the arrow is the chosen unit normal. For physical lengths, principal curvature and mean curvature have units of inverse length, while Gaussian curvature has units of inverse squared length. Rescaling a surface by $\lambda >0$ divides its principal curvatures by $\lambda$ and its Gaussian curvature by ${\lambda }^2$. Writing the saddle as $h=(x^2-y^2)/(2L)$ restores an explicit length scale: $K(0)=-1/L^2$.

The Theorema Egregium says that Gaussian curvature depends only on the first fundamental form and its first two coordinate derivatives. A local isometry therefore preserves Gaussian curvature, even if the shape operator and the principal curvatures change. ^[ref]

Here is the outline, rather than a full proof. The Gauss formula splits second derivatives of a parametric surface into tangential Christoffel symbols and the second fundamental form. The Christoffel symbols are determined by the first fundamental form and its first derivatives. Commuting the next partial derivatives gives the Gauss equation, which identifies $\det S$ with the intrinsic sectional curvature computed from the Riemann curvature tensor. Thus the apparently extrinsic product $k_1{k}_2$ is determined by the Riemannian metric. ^[ref]

One can see the distinction directly by rolling a plane strip with coordinates $(u,v)$ into the circular cylinder $X(u,v)=(R\cos (u/R),R\sin (u/R),v)$. Both have $I=d{u}^2+d{v}^2$, so the map is a local isometry, and is one-to-one on an open strip of width less than $2\pi R$. Extending $u$ over all real numbers wraps around periodically; it does not give a global isometry between the entire plane and the entire circular cylinder. A nonempty open patch of a sphere cannot be flattened by a local isometry into a plane, because its Gaussian curvature $1/R^2$ cannot become zero. The obstruction concerns preservation of the first fundamental form, rather than a particular drawing or choice of seam.

For perturbation $V_0{e}^{-{\beta }^2{x}^2}$ and oscillator inverse length $\alpha =\sqrt{m\omega /\hslash }$, direct Gaussian expectation values give $\Delta E_0=V_0\alpha /({\alpha }^2+{\beta }^2{)}^{1/2}$ and $\Delta E_1=V_0{\alpha }^3/({\alpha }^2+{\beta }^2{)}^{3/2}$. A narrow perturbation samples the odd state's node and suppresses its shift by two extra powers of $\alpha /\beta$. A broad perturbation approaches a constant energy offset for both states.

A gel band is a localized population of stained or otherwise detected molecules with sufficiently similar migration in one sample lane. One visible gel band need not contain only one nucleotide sequence. Two different linear DNA fragments of 375 base pairs can occupy the same unresolved gel band; equal migration supplies size evidence under suitable conditions, not a sequence determination.

Fluorescence from a bound stain can support an estimate of DNA quantity, but suitable staining and detector response matter.^[ref]

For a quantitative model with equal staining response per unit mass, background-subtracted, integrated fluorescence $I$ satisfies $I=km$. This requires an unsaturated detector and a common proportionality constant $k$; visual peak brightness alone does not supply $I$.

The amount of substance is $n=m/M$, where $M$ is molar mass. For double-stranded DNA with approximately equal average mass per base pair, $M$ is proportional to length $L$, so $n$ is proportional to $I/L$. In a constructed example, equal integrated signals from 250-base pair and 1000-base pair gel bands imply approximately equal masses but four times as many short DNA molecules. Conversely, equal numbers of those DNA molecules would give about four times the signal for the longer fragments in this model. Neither conclusion follows from darkness in a schematic drawing.

Separation of charged molecules through a gel in an electric field. Denaturing conditions suppress folded structures, helping compare RNA lengths; matching migration alone does not establish an identical nucleotide sequence or identical covalent bonds. In the ribozyme experiments, labelled RNA was detected after separation by exposing photographic film.

In DNA agarose gel electrophoresis, the backbone phosphate groups carry negative electric charge under the usual mildly alkaline buffer solution conditions. DNA therefore moves toward the positive anode, away from the negative cathode.^[ref]

The electric field points the other way: the electric force $\mathbf{F}=q\mathbf{E}$ reverses direction when $q<0$. The apparatus is driven by an external source of electric potential difference; the positive anode convention here belongs to that operating mode. Direction follows electric charge, so it is not a rule that all molecules move toward the same electrode.

The useful result is a spatial comparison: a mixture becomes separated gel bands, which can be compared with a DNA ladder or recovered for subsequent analysis. Agarose gel electrophoresis of linear double-stranded DNA, gel electrophoresis of denatured RNA, and SDS-PAGE of proteins use different conditions to make migration informative about chain size. An electric field supplies motion; the gel supplies a porous obstacle course.

A gel electrophoresis size calibration relates migration distance to known fragment lengths on the same gel. Here is an explicitly chosen model for linear double-stranded DNA, not measured data or a universal law. Let $L$ be the numerical length in base pairs and $d$ the numerical migration distance in millimetres from the wells:The logarithm has a dimensionless argument: $L/100$ is the length divided by 100 base pairs. The chosen DNA ladder fragments $L=100,250,500,1000$ give $d=55.00,50.22,46.61,43.00$, respectively. The gel bands therefore appear in the reverse order of length along the direction of migration.

To invert the calibration, rearrange the logarithm:Our chosen unknown $L=375$ gives $d=48.1116248\dots$, between the 250-base pair and 500-base pair references. Inserting that distance into the inverse recovers 375; retaining only $d=48.11$ gives about $375.12$ base pairs. The extra digits reflect rounding in this model, not the resolving power of a real gel.

For linear interpolation between those two reference gel bands, use the logarithm of length. Here the fraction of the distance from the 500-base pair reference toward the 250-base pair reference is $t={\log }_{10}(500/375)/{\log }_{10}2\approx 0.4150$, giving $L=500(250/500{)}^t=375$. Direct linear interpolation of lengths instead gives approximately $396.24$ base pairs. This difference explains why the choice of calibration coordinates matters. A real calibration must be checked over its usable range; folded RNA or circular DNA does not automatically share the linear double-stranded DNA calibration.

A porous gel changes the electrophoretic mobility of charged synthetic polymers through confinement, deformation and pore-to-pore dynamics. Free-draining charge and friction can both grow proportionally to chain length, leaving free-solution mobility nearly length independent; matrix obstruction adds size sensitivity. reptation supplies one weak-field long-chain regime, not a universal calibration. Matrix concentration, effective charge, solvent conditions, conformation, topology, adsorption, electro-osmotic drift and heating can all alter migration. Size standards should match the relevant chemistry and operating regime.

A geodesic is a curve whose tangent vector is parallel along itself under the Levi-Civita connection. With an affine parameter it has zero covariant acceleration:This is the Riemannian manifold version of moving straight ahead with constant speed. Its coordinate form is the geodesic equation. In Euclidean space it becomes $\ddot{\gamma }=0$, giving straight lines. On an embedded surface with its induced metric, the ordinary acceleration of such a curve is normal to the surface: all acceleration is used to follow the surface, rather than to turn within it.

On a unit round sphere, the great circle $\gamma (t)=(\cos t,\sin t,0)$ has $\ddot{\gamma }(t)=-\gamma (t)$, which is normal to the sphere. Two points at $t=0$ and $t=\pi /3$ can be joined along that circle by the short arc of length $\pi /3$ or the long arc of length $5\pi /3$. Each admits a constant-speed geodesic parametrization, yet only the short arc realizes their great-circle distance.

Every geodesic minimizes arc length on sufficiently short subarcs. A long segment can cease to be globally minimizing, as the sphere example shows; the first loss from a starting point is described by the cut locus. Thus “locally straight” and “shortest route between these endpoints” are distinct assertions. On a circular cylinder, different winding gives the same distinction, made explicit by Riemannian distance on a circular cylinder.

This provides a useful division of work. The geodesic equation predicts a path from an initial point and tangent vector; comparing arc lengths decides which candidate solves a global shortest-route problem. Variations of energy of a curve give the equation as an Euler-Lagrange equation, offering a proof outline and a connection with classical mechanics. ^[ref].

In coordinates $(\theta ,z)$ on a circular cylinder of radius $a$, the induced metric is $a^{2}d{\theta }^2+d{z}^2$. Its coefficients are constant, so the Christoffel symbols vanish and the affinely parametrized geodesic equation is ${\theta }^{\prime \prime }=z^{\prime \prime }=0$. The nonconstant solutions are generators when $c=0$, circular parallels when $d=0$, and helices when both are nonzero. Unrolling the cylinder with coordinate $u=a\theta$ makes all these curves straight lines in the Euclidean metric. Unit-speed parametrization additionally requires $a^2{c}^2+d^2=1$.

For a worked helix on a circular cylinder of radius $1$, choose $\theta (t)=2\pi t,z(t)=2t$ for $0\le t\le 1$. Unrolling gives the straight segment $(u,z)=(2\pi t,2t)$, so its arc length is $\sqrt{4{\pi }^2+4}$. Its acceleration in space points normally to the surface, consistent with the geodesic equation.

This geodesic is not globally shortest between its endpoints: they share the same angular position, and the vertical generator has length $2$. The helix minimizes length among paths with the same endpoints and one full winding, because their unrolled lifts join $(0,0)$ to $(2\pi ,2)$. In general, geodesics minimize sufficiently short subarcs; changing winding can produce a shorter route. Keeping this local/global distinction makes unrolling useful without mistaking every straight lift for an unrestricted shortest path.

A geometric spanner is a weighted graph on points in Euclidean space, with edge weights equal to their Euclidean distances, in which every pair has a graph path of length at most $t$ times its direct distance. Its stretch factor is the largest ratio $d_G(u,v)/\Vert u-v\Vert$, where $d_G$ is shortest-path distance. A square's four perimeter edges, without diagonals, give stretch factor $\sqrt{2}$: adjacent points have ratio one, and diagonal pairs have route length $2$ instead of $\sqrt{2}$. The aim is to retain short routes with fewer edges than a complete graph. ^[ref].

How much information about a shape do we need to keep? Euclidean geometry retains lengths and angles, allowing questions about the diagonal of a rectangle or the shortest route between two points. Topology retains the structure preserved by a homeomorphism, allowing questions about connectedness and whether a loop can shrink to a point, as recorded by the fundamental group. These are different levels of description of the same object.

For example, stretching a circle horizontally produces an ellipse. The stretching map and its inverse are continuous maps, so it gives a homeomorphism: the two curves have the same topology. It changes some Euclidean distances, however, so it is not an isometry. This distinction helps separate essential structure from a particular drawing, while explaining why measuring a shape requires more information than recognizing its topology.

The choice of geometry matters too. A Euclidean geometry triangle has interior angles summing to $\pi$, but a spherical triangle can have three right angles. Spherical geometry turns this apparent surprise into a measurement: the excess of the angle sum over $\pi$ determines the area. Begin with the Pythagorean theorem for flat measurements, then compare the spherical triangle below to see curvature change the rules.

Geophysical fluid dynamics studies fluid motion on rotating, stratified planets, including oceans, atmospheres, cores, and planetary interiors.

An initial current $U{1}_{|y|<a}$ with flat surface has two oppositely signed vortex sheets. On an f-plane, its local final geostrophic balance hasThe Rossby deformation radius controls the edge layers. The height is continuous and odd; the along-strip velocity is $-g{\eta }_f^{\prime }/f$ and has the sheet jumps. Outgoing inertia-gravity waves carry the energy not retained in this balanced component.

An initially resting, along-ridge-uniform shallow water layer with height disturbance ${\zeta }_0(x)$ conserves $q-f\zeta /h=-f{\zeta }_0/h$. Its balanced remainder satisfies the displayed modified Helmholtz equation, with $R_D=\sqrt{gh}/f$. On the infinite line,For each Fourier transform component, $\hat{\zeta }(t)={\hat{\zeta }}_s+({\hat{\zeta }}_0-{\hat{\zeta }}_s)\cos (\sqrt{f^2+gh{k}^2}t)$. The oscillatory part is inertia-gravity waves. For a localized Gaussian function on the infinite line those waves spread away, leaving the steady component locally; a reflecting finite basin generally retains oscillations.

A constant-depth layer initially at rest with a step in its free surface adjusts by radiating inertia-gravity waves. Conservation of linearized shallow-water potential vorticity leaves the bounded balanced profile ${\eta }_s={\eta }_0\mathrm{sgn}(x)(1-e^{-|x|/R_D})$, where $R_D=\sqrt{gH}/|f|$ is the Rossby deformation radius. The released potential energy per unit transverse length is $3\rho g{\eta }_0^2{R}_D/2$; the final kinetic energy is $\rho g{\eta }_0^2{R}_D/2$, leaving $\rho g{\eta }_0^2{R}_D$ in radiated waves.

The gravitational constant $G$ sets the strength of Newton's law of universal gravitation. Two point masses $M$ and $m$ a center-to-center distance $r$ apart attract with force magnitudeFor fixed masses and separation, doubling $G$ would double the force. For bodies with spherical symmetry, the same exterior formula follows from the spherical shell theorem; for arbitrary extended bodies, add the attractions of their mass density elements instead. The gravitational constant connects a laboratory attraction to the Newtonian gravitational field controlling a Kepler orbit.

The gravitational constant $G$ and local gravitational acceleration $g$ are different quantities. Applying Newton's second law to a test particle gives $g=F/m=GM/r^2$ for one spherical source. The source's mass and the observer's position change $g$; $G$ remains the same parameter of the law. A local falling-body experiment therefore measures a field, not $G$ alone. On rotating Earth, an instrument fixed to the surface also responds to centrifugal acceleration, while surrounding mass density and altitude change the gravitational contribution.

Dimensional analysis explains the units. Denote the base physical dimensions of length, mass and time by $\mathsf{L},\mathsf{M},\mathsf{T}$. Since force has dimension $\mathsf{ML}{\mathsf{T}}^{-2}$,By contrast, $[g]=\mathsf{L}{\mathsf{T}}^{-2}$ and the standard gravitational parameter $\mu =GM$ has dimension ${\mathsf{L}}^3{\mathsf{T}}^{-2}$. A small numerical value of a dimensional constant depends on the units chosen: it is the resulting force for specified physical masses and separation that conveys the laboratory scale.

NIST's current complete listing givesfor the 2022 CODATA adjustment, the latest adjustment listed when checked on 14 October 2026. Parentheses indicate a standard uncertainty $u(G)=0.00015\times 10^{-11}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, approximately $22$ parts per million relative to $G$. The adjustment year is 2022; NIST's May 2024 data update is a different date.^{[ref] [ref]}

For an original scale calculation, stipulate two $1\mathrm{kg}$ point masses separated by $1\mathrm{m}$ and insert this recommended $G$: $F=6.67430\times 10^{-11}\mathrm{N}$. The corresponding acceleration of either mass is $6.67430\times 10^{-11}\mathrm{m}{\mathrm{s}}^{-2}$ at that instant. Reducing the separation to $0.10\mathrm{m}$ multiplies both by $100$. These are ideal model predictions, not observations; their tiny scale explains why stray forces matter.

Being a constant of a physical law does not make the numerical value exact. The SI fixes the numerical values of seven defining constants, including the speed of light and Planck constant; $G$ is determined experimentally. Exact unit definitions still require physical realizations with measurement uncertainty: calibrating a length or a mass does not become error-free.^[ref]

A thin straight rod of length $l$, mass $m$, central moment of inertia $I=\beta m{l}^2$ and midpoint centre of mass starts beside a pointlike pole at that midpoint. An end impulse perpendicular to the rod gives translation $V=J/m$ and rotation $\Omega =J/(2\beta ml)$. At the first quarter-turn the trailing end is at horizontal distance $\pi \beta l-l/2$ from the pole. Strict clearance requires $\beta >1/(2\pi )$; equality is the grazing limit. The criterion is for free motion before contact, a negligibly thin rod and pole, and an impulse directed away from the pole.

For $V(x)=0$ on $0<x<a$ and $V(x)=V_w>0$ outside, the lowest bound state is even about $a/2$, has no nodes, and has $0<E<V_w$. Write $k=\sqrt{2mE}/\hslash$ and $\kappa =\sqrt{2m(V_w-E)}/\hslash$. The interior wavefunction is proportional to $\cos [k(x-a/2)]$; the exterior tails decay exponentially. Wavefunction matching at a finite potential step gives the displayed condition. Its lowest root has $0<ka<\pi$, so $E<{\pi }^2{\hslash }^2/(2m{a}^2)$, the infinite square well ground state energy. As $V_w$ increases, the penetration length $1/\kappa$ tends to zero and that limiting energy is approached from below.

On a cylinder with equal anyon-condensing boundaries, the two topological ground states split only through a virtual string implementing a logical operator. A perturbation of strength $h$ produces a splitting of order $J$ times a path-counting factor times $(|h|/J{)}^d$, where $d$ is the shortest supported logical-string length.