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On the graph of a function $z=f(x,y)$, assume $f$ is differentiable, its planar gradient $p$ is nonzero, and speed $U>0$ is measured by the three-dimensional Euclidean norm. The upward unit normal vector is $n=(-p,1)/\sqrt{1+|p{|}^2}$. Projecting the downward vertical vector $-e_z$ onto the tangent plane gives $-(p,|p{|}^2)/(1+|p{|}^2)$. By the Cauchy-Schwarz inequality, normalizing this projected vector produces the tangent direction with greatest height decrease at the prescribed speed, giving the displayed velocity. Its vertical component is $-U|p|/\sqrt{1+|p{|}^2}$, and its horizontal direction is parallel to $-\nabla f$. Consequently its planar paths agree with negative gradient flow up to a change of time, although the speeds differ. Where $p=0$, every tangent direction has zero first-order height change and this normalized formula is undefined.

In 1922, Otto Stern and Walther Gerlach used an electrically heated oven to emit neutral silver chemical atoms. Two platinum apertures, a pinhole and a slit $0.03$–$0.04\mathrm{mm}$ wide, supplied collimation. The beam passed along a $3.5\mathrm{cm}$ knife-edge pole in a nonuniform magnetic field. At its end, a glass plate collected the condensed silver. Continuous pumping maintained vacuum; the thin deposit required development even after eight hours.^[ref]

The field-off image established the undeflected slit width. A comparison put field-on and field-off deposits on the same plate. Microscopy revealed two deflected bands, no detectable central component, and broadening attributed to the velocity distribution. Alignment and distance from the knife-edge affected the pattern; precise magnetic field gradient measurements were still pending.^[ref]

The apertures act as a collimator: they constrain the initial transverse positions and velocities, so a magnetic displacement can be distinguished from an initially broad beam. Vacuum reduces collisions that would destroy this preparation. Deposition integrates arrivals over the exposure; a darkened band is not a time-resolved record of individual chemical atoms. The field-off comparison tests whether the magnet changes the spatial distribution, while alignment checks address a geometric source of apparent asymmetry. Finite sensitivity qualifies the absence of a central deposit: it is an experimental upper limit, not a proof that exactly zero atoms landed there.

For an equally populated ideal pair of angular momentum eigenstates, the magnetic moment component takes two values, $\pm {\mu }_B$, and the displacements at fixed velocity are $\pm d(v)$. Compare a specified classical alternative: every chemical atom carries a permanent magnetic moment of fixed magnitude ${\mu }_B$, with independently isotropic initial orientations and no dissipative alignment. If $u=\cos \theta$, the sphere's area element is $2\pi du$, so $u$ has a uniform distribution on $[-1,1]$. With precession preserving this component in the ideal geometry, the displacement $d(v)u$ fills the interval $[-d(v),d(v)]$ continuously. A pair of bands therefore tests allowed components rather than merely proving that silver is magnetic. Other classical preparation assumptions would be different models and must be stated.

To include finite instrumental width, choose a normalized Gaussian distributionwith standard deviation $\sigma =0.035\mathrm{mm}$, representing a combined field-off spatial width. At fixed velocity the model probability density functions areEach has unit integral and zero mean; the different shapes, rather than a different average displacement, distinguish them. The discrete model has variance ${\sigma }^2+d(v{)}^2$, whereas the classical model has variance ${\sigma }^2+d(v{)}^2/3$. Their common mean also explains why the raw average landing position is insufficient to identify the internal quantum state.

Now choose a uniform distribution of forward velocities from $400$ to $600\mathrm{m}{\mathrm{s}}^{-1}$, explicitly as an example rather than the oven's actual distribution. Average either conditional probability density function over that interval. The positive-channel displacement ranges from $0.1776$ to $0.3995\mathrm{mm}$, since slower chemical atoms spend longer accelerating and drifting. The two components broaden into bands even though their internal magnetic moment values remain discrete. The corresponding classical profile continues to populate the center. The displayed strip darkness is proportional to predicted deposited number density, with equal total arrivals; it is not a photograph or a calibrated optical response of a developed film.

Increasing the spatial width until it is comparable with $d$ can hide the two maxima. At fixed velocity, differentiating the two-state probability density function twice gives $p_{\mathrm{two}}^{\prime \prime }(0)={\varphi }_{\sigma }(d)(d^2-{\sigma }^2)/{\sigma }^4$: the central minimum becomes a central maximum when $\sigma >d$. Conversely, a broad distribution of velocities need not create a new internal component: it spreads each existing component through the $v^{-2}$ dependence. Further instrumental limits include collisions, aperture clipping of strongly deflected trajectories, asymmetric pole geometry, and nonlinear film development. Field-off widths and a mapped magnetic field gradient are needed to separate such effects. For a modern counting experiment, channel calibration and efficiency corrections belong to Stern-Gerlach expectation estimation; spatial separation alone does not supply those statistics.

Let the atomic beam travel along $x$, with velocity $v>0$, and let $z$ measure displacement along the dominant magnetic field gradient. For a fixed effective magnetic moment component ${\mu }_z$, the magnetic dipole potential energy is $U\simeq -{\mu }_z{B}_z$. Taking its negative partial derivative givesA spatially uniform magnetic field can produce torque and Larmor precession, but cannot produce this translational separation. In a quantum state following a local internal energy eigenvalue $E_n(B)$, the more general expression is $F_z=-{\partial }_z{E}_n$, with effective magnetic moment $-\partial E_n/\partial B$. Treating that moment as constant is an approximation with a physical regime, not an extra measured fact. Assume the local magnetic field direction stays nearly fixed across the sampled beam, and that internal states follow it without transitions, as in the adiabatic theorem. The scalar gradient model describes the beam's neighborhood; it omits the full magnetic field's transverse components and fringe regions.

Assume a constant $G$ over magnet length $\ell$, followed by a field-free drift $D$. Neglect transverse motion at entry and changes in longitudinal velocity. Newton's second law gives transverse acceleration $a_z={\mu }_{z}G/M$ for whole-atom mass $M$. The magnet transit time is $t_m=\ell /v$, and the drift time is $t_d=D/v$. The magnet supplies both a displacement and an exit transverse velocity:The $\ell D$ contribution is motion with the acquired transverse velocity, not continued acceleration outside the magnet. For the plate-at-exit geometry, set $D=0$. For a nonzero entrance transverse velocity $w$, add $w(\ell +D)/v$. These terms show why initial collimation and flight geometry matter when inferring a magnetic moment from a deposit.

Choose hypothetical parameters $M=1.79\times 10^{-25}\mathrm{kg}$, $G=300\mathrm{T}{\mathrm{m}}^{-1}$, $\ell =0.035\mathrm{m}$, $D=0.100\mathrm{m}$, and $v=500\mathrm{m}{\mathrm{s}}^{-1}$. Approximate the two effective magnetic moments by ${\mu }_z=\pm {\mu }_B$, where the Bohr magneton is ${\mu }_B\simeq 9.274\times 10^{-24}\mathrm{J}{\mathrm{T}}^{-1}$. Then $|a_z|\simeq 1.55\times 10^4\mathrm{m}{\mathrm{s}}^{-2}$, $t_m=70\mu \mathrm{s}$ and $t_d=200\mu \mathrm{s}$. The positive displacement is about $0.0381\mathrm{mm}$ inside the magnet plus $0.2176\mathrm{mm}$ during drift: $d=0.2557\mathrm{mm}$, with full separation $2d=0.5114\mathrm{mm}$. These chosen values illustrate the model and are not reconstructed 1922 measurements.

The exit transverse velocity is only about $1.09\mathrm{m}{\mathrm{s}}^{-1}$, small compared with the chosen forward velocity; this checks the small-deflection assumption. Halving $G$ halves $d$, doubling $v$ divides $d$ by four, and setting $D=0$ removes most of this example's separation. A real pole has fringe magnetic fields and a position-dependent gradient; these require a more complete trajectory calculation.

A residual accuracy level in a bound for stochastic gradient descent with a fixed step size and nonvanishing gradient noise. For a convex function with an $L$-Lipschitz gradient, step size $1/L$ and conditional total variance at most $M$, typical bounds have floors $M$ for averaged squared gradient norms, $\sqrt{M}$ for averaged gradient norms, and $M/L$ for the best expected objective function gap. Such floors are upper bounds depending on the analysis and assumptions, rather than claims that these exact values are attained or that every last iterate converges.

Target: gradient-descent.bigb, under Stochastic gradient descent.

For an unbiased stochastic gradient $H$ at $X$ with finite conditional second moment, expand $H=\nabla f(X)+(H-\nabla f(X))$ in the squared Euclidean norm. The cross term has zero conditional expectation, which proves the displayed identity. It separates the squared true gradient from the total conditional variance of its random estimate.

Target: gradient-descent.bigb, under Stochastic gradient descent.

For a differentiable strongly convex function with modulus $\gamma >0$ and global minimizer $x^{\ast }$, the supporting inequality givesCompleting the square shows that the last two terms are at least $-\Vert \nabla f(x){\Vert }^2/(2\gamma )$, which proves the result. Combining this with the descent lemma converts squared gradient decrease into a contraction of the objective function gap.

Target: mathematical-optimization.bigb, under Line search.

Consider an objective function that is a quadratic polynomial, with a singular optimization constraint:Its only feasible point is $x_{\ast }=0$, giving $p_{\ast }=9$. For $\lambda \ge 0$, completing the square yieldsEvery finite Lagrange multiplier gives $q(\lambda )<9$, but $q(\lambda )\to 9$ as $\lambda \to \infty$. Thus $d_{\ast }=p_{\ast }=9$: strong duality holds, and the primal optimum is attained, yet dual attainment fails. The inner minimizers $3/(1+\lambda )$ approach $0$ without ever becoming feasible at finite $\lambda$.

Both the objective function and the constraint function are convex functions. However, Slater's condition fails because $x^2<0$ is impossible. At the optimum, stationarity would require $-6+\lambda \cdot 0=0$, which no finite Lagrange multiplier satisfies. Compare KKT regularity at a local minimum: a zero gradient of an active constraint can prevent a multiplier certificate. This example has zero duality gap without the Strong Lagrangian property.

For a locally uniformly elliptic operator $L=a^{ij}{D}_{ij}+b^i{D}_i+c$ with locally bounded coefficients and $c\le 0$, a $C^2$ function satisfying $Lu\ge 0$ on a connected domain cannot attain a nonnegative global maximum inside unless it is constant. If $c=0$, the maximum can have either sign. If the maximum set were proper, choose an interior ball in its complement tangent to that set. The Hopf boundary point lemma gives a nonzero normal derivative at the contact point, contradicting the vanishing gradient at an interior maximum.

The strong minimum principle for elliptic operators follows by applying this to $-u$. For nonnegative $u$ with $Lu\le 0$, replacing $c$ by $\min (c,0)$ retains the inequality, so the strict positivity-or-zero conclusion does not require an original sign restriction on $c$.

For a smoothed image signal $u_{\sigma }$, average the gradient outer product as $J_{\rho }=G_{\rho }\ast (\nabla u_{\sigma }\nabla u_{\sigma }^T)$. The dominant eigenvector estimates the normal to a coherent image edge, and eigenvalue contrast measures local directional structure. Two distinct scales separate differentiation from orientation averaging.

The BV lattice inequality $|D\min (v,w)|+|D\max (v,w)|\le |Dv|+|Dw|$ applied to two indicator functions gives the perimeter inequality. For smooth approximations, minima and maxima partition the two gradients; sequential lower semicontinuity passes the estimate to BV. It prevents a crossing of minimizers with strictly ordered set forcing.

Superfluidity is a quantum-fluid response with phase stiffness and flow without viscous dissipation in an appropriate low-speed regime. A superfluid supports a phase-gradient current; defects and excitations can limit that flow. Bose-Einstein condensation concerns macroscopic one-particle occupation and is a distinct criterion: the Gaussian phase stiffness theory can have divergent absolute-phase fluctuations while retaining a cost for gradients.

Why does cooling a liquid make it flow more freely through a tiny gap? Helium-4 supplies an experimental entrance to this question: the helium slit-flow experiment compares the motion of liquid levels on opposite sides of the transition into superfluidity. A flow measurement probes resistance to motion; a measurement of Bose-Einstein condensation probes occupation of a quantum state. The two-fluid model of superfluidity and the Landau criterion connect these questions to entropy transport and to the energy required to create an excitation.

For a condensate $\psi =\sqrt{n}{e}^{iS}$ with kinetic operator $-{\hslash }^2{\nabla }^2/(2m)$, its particle current is $n\mathbf{u}$ with the displayed velocity. This follows by subtracting the Gross–Pitaevskii equation from its complex conjugate. More generally, in units where the kinetic operator is $-D{\nabla }^2$, the current velocity is $2D\nabla S$. The complex argument gradient and current velocity coincide only when $D=1/2$.

For an embedded surface with unit normal vector $\mathbf{n}$, the surface gradient is ${\nabla }_{s}f=(\mathbf{I}-\mathbf{n}{\mathbf{n}}^T)\nabla f$. It projects the ambient gradient onto the tangent space and is independent of how $f$ is extended normally off the surface.

Surrogate Hamiltonian Monte Carlo uses an auxiliary smooth probability density function $\nu$ to generate trajectories but applies a Metropolis–Hastings acceptance probability using the intended target $\mu$. For Gaussian momenta and an involutive Metropolis proposal generated by flipped leapfrog integration, the acceptance probability is $\min (1,\mu (x^{\prime })e^{-\Vert p^{\prime }{\Vert }^2/2}/[\mu (x)e^{-\Vert p{\Vert }^2/2}])$. Only the surrogate gradient is required during the trajectory; target density evaluations are still required at the endpoints.

For a symmetric positive-definite matrix $A$, solving $A{x}_{\ast }=b$ is equivalent to minimizing the quadratic form with linear termWriting the error vector $e=x-x_{\ast }$ and expanding gives the exact energy norm identityThe cross term vanishes because $A{x}_{\ast }=b$. With $\eta =Se$ and $B=S^{-1}A{S}^{-1}$, the same quantity is ${\eta }^{T}B\eta /2$. Symmetric positive-definite preconditioning changes coordinates without changing the energy assigned to a corresponding error vector.

In an orthonormal eigenbasis of $A$, the level set $e^{T}Ae=t>0$ has semiaxis lengths $\sqrt{t/{\lambda }_i}$. Its longest-to-shortest axis ratio is $\sqrt{{\kappa }_2(A)}$. A large spectral condition number of a positive-definite matrix therefore appears geometrically as a thin ellipsoid. Good scaling makes the transformed ellipsoid rounder, allowing a simple search direction to improve all coordinates together.

The negative gradient of $F$ is the linear-system residual $r$. Relative to the norm $\Vert v{\Vert }_M=\sqrt{v^{T}Mv}$, a steepest direction is $z=M^{-1}r$: the Cauchy-Schwarz inequality applied to $r^{T}v=(S^{-1}r{)}^T(Sv)$ shows that the greatest decrease per unit $M$-length occurs along this direction. Equivalently, ordinary gradient descent in $y=Sx$ becomes preconditioned steepest descent in $x$.^[ref]

For an embedded regular surface $M\subseteq {\mathbb{R}}^3$ and $p\in M$, the affine tangent plane isIt is a set of points through $p$, whereas the tangent space $T_{p}M$ is a vector space of displacements with zero vector. Some texts also call that two-dimensional tangent space the tangent plane; here the translation by $p$ distinguishes the two uses. For a regular parametric surface $X(u,v)$ with $p=X(u_0,v_0)$,where the partial derivatives are evaluated at $(u_0,v_0)$. Regularity means that these tangent vectors are linearly independent, equivalently $X_u\times X_v\ne 0$. Their cross product is a normal vector. ^[ref]

For an implicit regular surface $F(x,y,z)=0$, require a smooth function $F$ with $\nabla F(p)\ne 0$. ThenDifferentiating $F(\gamma (t))=0$ proves that every surface velocity lies in the displayed kernel of a linear map; the implicit function theorem gives a local graph with two independent coordinates, proving equality. The gradient is the normal vector with respect to the ambient dot product. ^[ref]

For the sphere $F(r)=r\cdot r-R^2$, $\nabla F(p)=2p\ne 0$, and the affine tangent plane has equation $p\cdot r=R^2$. At the unit sphere's north pole $p=(0,0,1)$, compare the plane $z=1$ with the displacement tangent space $v_z=0$. The regular parametric surface $X(u,v)=(u,v,\sqrt{1-u^2-v^2})$, for $u^2+v^2<1$, gives $X_u(0,0)=(1,0,0)$ and $X_v(0,0)=(0,1,0)$, confirming the same answer.

This analytical illustration shows a unit sphere and a vertical cross-section. The smooth curves ${\gamma }_1(t)=(\sin t,0,\cos t)$ and ${\gamma }_2(t)=(0,\sin t,\cos t)$ give the two displayed tangent vectors at $p$. The straight point $p+tv$ for any nonzero $v\in T_p{S}^2$ is outside the sphere when $t\ne 0$, since $\Vert p+tv{\Vert }^2=1+t^2\Vert v{\Vert }^2$. A tangent plane therefore captures first-order motion, not an exact patch of the regular surface.

The same idea makes local approximation useful. For the graph of a function $z=f(x,y)$ with $f$ a smooth function, the affine tangent plane is $z=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)$. ^[ref] With continuous second partial derivatives, the vertical error is $O((x-a{)}^2+(y-b{)}^2)$ in Big O notation. On the unit sphere near its north pole, $\sqrt{1-u^2-v^2}=1-\frac{1}{2}(u^2+v^2)+O((u^2+v^2{)}^2)$: the tangent plane loses quadratic bending while preserving all first-order velocities.

A failed regularity test needs interpretation. Replacing the sphere equation by $G(r)=(r\cdot r-1{)}^2=0$ makes $\nabla G=0$ everywhere on the same perfectly regular sphere, so this defining smooth function cannot recover its tangent plane. Similarly, the usual spherical parametric surface loses matrix rank at the poles, while the graph parametrization above remains regular there. A vanishing gradient or degenerate parametrization invalidates that calculation; it alone does not prove that the underlying geometry has no tangent plane.

For a level set $G=0$, the tangent plane formula $\nabla G(p)\cdot (x-p)=0$ requires both that $p$ belongs to the surface and that its gradient is nonzero. Evaluating the gradient at an off-surface point instead gives a plane tangent to the level set through that different point. For example $G=x^{2}y-4z{e}^{x+y}+11$ has $G(2,-2,9)=-33$, so the gradient plane there cannot be a tangent plane to $G=0$. Holding the first two coordinates fixed gives the on-surface point $(2,-2,3/4)$ and its plane $11x-y+4z=27$.

Subtracting the quadratic Taylor polynomial at the rescaling centre makes the function, gradient and Hessian matrix vanish there. Dividing by $A{r}^{2+\mu }$ preserves the normalized Hölder oscillation of the Hessian matrix while turning a small forcing-to-oscillation ratio into a harmonic limiting equation. The three vanished Taylor coefficients provide local bounds needed by the Arzelà-Ascoli theorem.

The temperature gradient is the gradient $\nabla T$ of the temperature field. It points in the direction of fastest temperature increase.