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A practical choice is a BFGS method, using exact gradients and a line search. At dimension $d=100$, its $d$-by-$d$ inverse-Hessian matrix approximation is small, whereas forming a million rank-one contributions to the exact Hessian matrix is expensive. If $F(x)={\sum }_i{e}^{\Vert x-a_i{\Vert }^2}$, the explicit derivatives areThe positive-definite Hessian matrix and coercivity show that the objective has one global minimizer.

For numerical stability one can minimize $\Phi =\log F$ instead, which has exactly the same minimizer. Put $r_i=\Vert x-a_i{\Vert }^2$, $w_i=e^{r_i}/{\sum }_j{e}^{r_j}$, and $\bar{a}={\sum }_i{w}_i{a}_i$. Then the logarithmic gradient of an exponential squared-distance sum givesEvaluate the weights as $e^{r_i-r_{\max }}/{\sum }_j{e}^{r_j-r_{\max }}$ to avoid floating-point overflow. A quasi-Newton method step iswhere $H_n$ is the positive-definite inverse-Hessian matrix approximation updated by BFGS, and ${\alpha }_n$ is chosen by a line search, for example one satisfying the Wolfe conditions. Under the standard smoothness and nondegeneracy assumptions at this strongly convex minimizer, full-memory BFGS has local superlinear convergence. An exact Newton's method in optimization would have local quadratic convergence, but pays for those much more expensive exact second derivatives.^[ref]

Writing $g=e^{x^{\mathsf{T}}x}$, the gradient and Hessian matrix areConsequently, for any nonzero $v$,In particular the Hessian matrix is a positive semidefinite matrix everywhere, so the Hessian criterion for convexity proves $g$ is convex. The strict inequality along every nonconstant line also proves strict convexity.

The derivative is represented by the gradient, and the Hessian matrix contains the second partial derivatives:Equivalently the derivative as a linear map is $Df(x,y)(u,v)=(4x^3-4y)u+(4y-4x)v$.

Assume $f$ and $g$ are continuously differentiable near the point. At a regular feasible point, meaning $g(x)=0$ and $\nabla g(x)\ne 0$, a differentiable constrained local extremum satisfies the Lagrange multiplier equationsThey arise by differentiating the optimization Lagrangian $L(x,\lambda )=f(x)-\lambda g(x)$ with respect to all coordinates and the multiplier. Geometrically, the objective has zero derivative along the tangent space of the constraint surface, so its gradient must lie in the normal direction.

Solve these equations for candidates, examine feasible points where the regularity condition fails separately, and compare objective values or use a second-order test restricted to feasible directions. The equations locate stationary points; they do not by themselves establish which are minima or whether a global minimum is attained.

The full, undamped Newton's method in optimization step isThe eigenvector identity gives $A^{-1}b=2b$. Consequently $x_1=-b$ for every starting point. This is a one-step Newton solution of a quadratic stationary equation: its gradient is an affine function, so a single exact Newton solve resolves it completely.

Use compatible dimensions $A\in {\mathbb{R}}^{d\times d}$ and $b,x\in {\mathbb{R}}^d$; the printed “b-by-b” size is read consistently with the displayed products and domain. Take derivatives as column gradients. Symmetry of $A$ givesFor example, the directional differential of $x^{T}Ax$ is $d{x}^{T}Ax+x^{T}Adx=2(Ax{)}^{T}dx$. The linear term contributes $b$ to the gradient and zero to the Hessian matrix.

The gradient vanishes at $0$. Decompose any vector as $x=tb+w$, with $w\perp b$. Symmetry of $A$ and $Ab=\frac{1}{2}b$ imply $b^{T}Aw=(Ab{)}^{T}w=0$. Since $\Vert b\Vert =1$,If $d\ge 2$, choose a nonzero $w\perp b$. Arbitrarily close to zero, the direction $b$ gives negative values and the direction $w$ gives positive values. Thus $0$ is a saddle point for $d\ge 2$. Equivalently, the Hessian matrix has one negative eigenvalue and all its other eigenvalues positive.

The dimension matters: if $d=1$, the orthogonal complement is zero-dimensional and $g(tb)=-t^2/2$. In that case $0$ is the unique global maximum. This edge case is not excluded by the stated hypotheses. The decomposition is a rank-one downdate along an eigenvector.

At $x=0$, the gradient from the preceding part vanishes and the Hessian matrix is zero. The second-order test is therefore inconclusive. Instead inspect the eigenline:For arbitrarily small positive and negative $t$, these values have opposite signs. Thus $0$ is neither a local maximum nor a local minimum. In dimension at least two it is a degenerate saddle point; in one dimension it is a stationary inflection point. Looking only at the zero Hessian matrix would not determine this classification.

Gradient descent uses only the current gradient and a suitable step size; its additional storage is order $d$. It is attractive for very large $d$ or when low per-iteration cost matters most. For a smooth strongly convex function, standard step choices give linear convergence, with a rate sensitive to the condition number. Mere convexity gives a global-minimum interpretation but does not alone imply a linear rate; for nonconvex objectives the usual guarantee concerns stationary points.

BFGS uses successive gradients to update an approximation to inverse curvature, incorporating information that can correct poor scaling. Full BFGS uses order $d^2$ storage and update work. With suitable line searches and local smoothness and positive-curvature assumptions near a nondegenerate minimum, it has superlinear convergence. This often favours it in moderate dimensions when gradients are not extremely cheap relative to the extra work.

Neither gradient descent nor BFGS requires an explicitly evaluated Hessian matrix. Thus an expensive Hessian matrix is compatible with both, rather than a reason to reject BFGS. If an accurate Hessian matrix and its factorization are exceptionally cheap, Newton's method in optimization becomes an additional reasonable option. Convexity helps identify a stationary point as a global minimum; without it, both methods need suitable descent and curvature safeguards. The BFGS update learns curvature from the Quasi-Newton secant equation.^[ref]

A proton-precession magnetometer measures the scalar magnitude $F=|\mathbf{B}|$ of the local magnetic field. Its sensor contains a proton-rich liquid, such as a hydrocarbon or suitable water-based fluid. A strong current through a coil briefly polarizes the nuclear magnetic moments, producing a small net magnetization. The polarizing field is directed sufficiently across the local magnetic field to leave a transverse component when the current is switched off.

The protons then undergo Larmor precession about the ambient magnetic field. Their rotating net magnetization induces a decaying alternating voltage in a receiving coil. Measuring the frequency determines the field:For example, $50\mu \mathrm{T}$ gives about $2.13\mathrm{kHz}$. The known proton gyromagnetic ratio provides an absolute frequency-to-field scale; this instrument does not measure the field's direction. Strong spatial gradients cause different parts of the liquid to precess at different rates and shorten the usable signal, which matters very close to a buried steel object.^[ref]

External chemical signals can guide cell migration, but the statement is too strong if “determined” means that they uniquely specify whether, how fast and in which direction a cell moves. A chemoattractant gradient biases a cell's internal polarity machinery; it does not directly supply the mechanical work of locomotion. Chemotaxis is directional bias, whereas a change in speed or persistence without directional preference is chemokinesis.

A chemical response depends on receptor abundance, signalling adaptation and the cell's existing cell polarity. The same external field can therefore yield different trajectories in different cell states. Uniform stimulation can increase motility without providing a direction, and spontaneous fluctuations of the actin system can initiate movement without a spatial chemical instruction. Persistent migration combines feedback within the cell with a cue that biases that feedback.

Mechanical and geometrical inputs also matter. Durotaxis describes a bias associated with substrate stiffness, while haptotaxis follows spatial variation in adhesive ligands. Contact guidance by fibre alignment can channel motion even when soluble signals are uniform. Narrow spaces constrain the available paths and can force a switch of protrusive mechanism. These effects are not wholly separate from chemistry: cells sense mechanical forces through molecular interactions. Nevertheless, their directional information resides in mechanics or geometry, not necessarily in an imposed soluble concentration gradient.

Contacts between cells supply another source of organization. Experiments on neural crest cells showed that collisions can cause protrusion collapse and repolarization, and that perturbing noncanonical Wnt signalling disrupts this behaviour and group directionality.^[ref] Contact inhibition of locomotion therefore helps generate collective trajectories through local interactions, rather than each cell independently reading a complete external chemical map. Cohesion and other attractive interactions can prevent such repulsion from simply dispersing the group.

A discriminating experiment independently varies soluble gradient, adhesive pattern and mechanical confinement. Measure direction, speed and persistence separately; reverse the gradient, remove it, and then block the relevant receptor while checking viability and the ability to move. If a directional bias follows the reversed chemical field, that supports chemical guidance. If speed rises under uniform stimulation, that supports chemokinesis. If trajectories remain aligned with fibres after chemical reversal, physical guidance is influential. Manipulating intracellular polarity can test whether a cue is instructive or merely permissive.

The useful conclusion is that motion emerges from an internally powered, state-dependent cell interacting with chemical, mechanical and cellular surroundings. External chemical signals are often decisive inputs, but the trajectory is a property of the coupled system.

The Paul trap alternates focusing and defocusing electric forces. Their time average along a moving trajectory need not vanish, even when the electric field averages to zero at every fixed point. A spatial gradient makes the fast displacement correlate with the driving electric field; the RF-averaged trapping potential calculates the resulting restoring force.

Wolfgang Paul's lecture describes hyperbolic ring/endcap electrodes, loading by electron-impact ionization of low-pressure gas, and resonant ejection through an endcap hole into an electron multiplier. Sweeping RF voltage produces an air mass spectrum.^[ref]

In a fixed-temperature convection problem, the boundary temperature is prescribed, and its perturbation satisfies a thermal Dirichlet boundary condition. A homogeneous condition means the boundary follows the prescribed conductive background temperature. This does not necessarily mean a vertical wall has a spatially constant temperature; that additional assumption can conflict with a background vertical gradient.

For an incompressible phase, this coefficient compares its actual axial momentum flux with the uniform-profile flux at the same intrinsic mean velocity. Cauchy-Schwarz gives $D_1\ge 1$ where the mean is nonzero. A spatially varying profile can have a larger second moment even when its mean speed is unchanged.

With constant $D_1$, expansion of ${\partial }_t(\alpha u_1)+{\partial }_x(\alpha D_1{u}_1^2)$ and use of ${\alpha }_t+(\alpha u_1{)}_x=0$ gives $\alpha (u_1{)}_t+(2D_1-1)\alpha u_1(u_1{)}_x+(D_1-1)u_1^2{\alpha }_x$. Thus a profile correction changes both convection and phase-fraction-gradient coupling in the averaged momentum equation. If $D_1$ varies, an extra $\alpha u_1^2(D_1{)}_x$ remains; a zero mean should be treated through the undivided second moment rather than an undefined coefficient.

A plastic flow rule prescribes the direction and rate of plastic straining once a yield condition is active. In a plastic-potential model its direction is the stress gradient of $g$, multiplied by a nonnegative plastic multiplier. An associated flow rule takes $g$ equal to the yield function. The stress-strain split, loading conditions and hardening evolution are needed along with this equation to close an elastoplastic model.

On a bounded connected Lipschitz domain $U$,for every $u\in H^1(U)$. If this failed, the Rellich-Kondrachov compactness theorem would produce a normalized strong $L^2$ limit with zero gradient, hence a constant function; continuity of the Sobolev trace theorem would force that constant to vanish, contradicting its unit norm.

In three dimensions let $f\in L^{6/5}({\mathbb{R}}^3)$ have compact support. Let $v_R\in H_0^1(B_R)$ solve $-\Delta v_R=f$ and extend it by zero. There is a unique $v\in {\dot{H}}^1({\mathbb{R}}^3)$ with $\int \nabla v\cdot \nabla \varphi =\int f\varphi$ for all $\varphi \in {\dot{H}}^1$. Indeed Holder inequality and the homogeneous Sobolev inequality make the right side a continuous linear functional in the gradient norm, and the Riesz representation theorem applies to this Hilbert space.

The weak formulation says that $v_R$ is the orthogonal projection of $v$ onto the closed subspace consisting of zero extensions of $H_0^1(B_R)$. These subspaces increase with $R$ and their union contains every test function, which is dense by the definition of the homogeneous Sobolev space. The best-approximation property of orthogonal projection therefore gives $\Vert \nabla (v_R-v){\Vert }_2\to 0$. The homogeneous Sobolev inequality also gives $\Vert v_R-v{\Vert }_6\to 0$. This convergence does not generally extend to the inhomogeneous Sobolev space $H^1$, as the spherical mean of a Poisson solution outside its source shows for a source of nonzero total mass.

Split the units into visible units $v$, whose patterns occur in the training data, and hidden units $z$. Let $q(v)$ be the empirical probability distribution and maximize the average log-likelihood $\ell ={\sum }_{v}q(v)\log p_{\beta }(v)$, where $p_{\beta }(v)={\sum }_z{p}_{\beta }(v,z)$. The positive phase clamps $v$ to data and averages over $q(v)p_{\beta }(z\mid v)$; the negative phase releases all units and averages over $p_{\beta }(v,z)$. The displayed difference uses these two expectations. When both units are visible, the positive pair product is supplied directly by the data.

For a proof, write $\log p_{\beta }(v)=\log {\sum }_z{e}^{-\beta E(v,z)}-\log Z$. Differentiating each finite sum gives, respectively, $-\beta ⟨\partial E{⟩}_{z\mid v}$ and $-\beta ⟨\partial E{⟩}_{v,z}$. Since $\partial E/\partial w_{ij}=-x_i{x}_j$, subtraction proves the rule. Similarly ${\partial }_{{\theta }_i}\ell =-\beta (⟨x_i{⟩}_{+}-⟨x_i{⟩}_{-})$. A small positive step along these gradients increases log-likelihood. The negative term suppresses associations overproduced by the model; strengthening observed pairs alone misses this correction. The identity requires equilibrium expectations; estimates from insufficiently mixed sampling can give a biased update.

Potential energy assigns energy to a system's arrangement, so a change of arrangement accounts for the work done by a conservative force. For a time-independent potential energy $V(\mathbf{r})$,The gradient points toward increasing potential energy; the force points toward decreasing potential energy. Only differences matter: adding a constant to $V$ changes neither the force nor its work. This endpoint rule follows by substituting $-\nabla V$ into the line integral for work. ^[ref]

Near Earth's surface, stipulate constant gravitational acceleration $g=9.81m/{\mathrm{s}}^2$ and measure height $h$ upward from a chosen zero. Then gravitational potential energy is $V=mgh$. Raising a $2\mathrm{kg}$ mass by $0.5\mathrm{m}$ increases its gravitational potential energy by $9.81\mathrm{J}$, while gravity does $-9.81\mathrm{J}$ of work. These are computed model values, not measurements of a particular lift. An ideal spring obeying Hooke's law instead has $V=k{x}^2/2$, with displacement $x$ from its relaxed state and spring constant $k$: $k=100N/m$ and $x=0.2\mathrm{m}$ give $V=2\mathrm{J}$ relative to $x=0$. ^[ref]

A potential function $V$ assigns a scalar to each point of a configuration or spatial domain. In mechanics it represents potential energy, and its negative gradient gives the corresponding conservative force.