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Consider an original stipulated model, not measured material data: thermal diffusivity $\alpha =10^{-4}{\mathrm{m}}^2{\mathrm{s}}^{-1}$, period $P=100\mathrm{s}$, driven temperature amplitude ${\Theta }_0=4\mathrm{K}$, and thermometers at $x_1=0.020\mathrm{m}$ and $x_2=0.050\mathrm{m}$. Without side losses, $a=b=17.7245{\mathrm{m}}^{-1}$ and thermal penetration depth $\delta =0.05642\mathrm{m}$. ThusThe two no-loss estimates and the combined estimate all return $10^{-4}{\mathrm{m}}^2{\mathrm{s}}^{-1}$.

Now retain the same thermal diffusivity and drive but introduce $\lambda =0.010{\mathrm{s}}^{-1}$. The exact complex wavenumber gives $a=19.1863{\mathrm{m}}^{-1}$ and $b=16.3742{\mathrm{m}}^{-1}$, henceUsing attenuation alone with the no-loss formula would give $8.5343\times 10^{-5}{\mathrm{m}}^2{\mathrm{s}}^{-1}$; using lag alone would give $1.17174\times 10^{-4}{\mathrm{m}}^2{\mathrm{s}}^{-1}$. Their disagreement is informative. If an independent volumetric heat capacity measurement gives $\rho c_p=2.0\times 10^6\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}$, the combined estimate implies thermal conductivity $k=200\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$ and $H=2.0\times 10^4\mathrm{W}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}$.

The exact complex amplitudes can be checked independently by discretizing $\alpha {\hat{\theta }}^{\prime \prime }-(\lambda +i\omega )\hat{\theta }=0$ and solving the resulting linear system with a tridiagonal matrix. Halving the spatial step quarters the discrepancy, as expected for centered finite differences. Separately fitting the real time traces to a constant plus sine and cosine terms recovers both amplitudes and their lag; this also checks the sign convention used in the combined estimate.

The scaled backward finite difference uses the current and preceding nodal values to approximate a first derivative. If a function has two continuous derivatives, Taylor's theorem gives $D_h^{-}u(x)=u^{\prime }(x)-h{u}^{\prime \prime }(x)/2+o(h)$ as $h\to 0$. Composition with the scaled forward difference operator gives $D_h^{+}{D}_h^{-}u(x)=(u(x+h)-2u(x)+u(x-h))/h^2$, the second-order central difference for the second derivative.

The Bernoulli polynomials extend the Bernoulli numbers by inserting the exponential function $e^{xt}$ into their exponential generating function:For example, $B_0(x)=1$, $B_1(x)=x-1/2$, $B_2(x)=x^2-x+1/6$, and $B_3(x)=x^3-3x^2/2+x/2$.^[ref]

Their crucial property is a finite difference identity:Indeed, subtracting the two exponential generating functions cancels $e^t-1$ and leaves $t{e}^{xt}$. Thus $B_{p+1}(x)/(p+1)$ plays for discrete summation the role that $x^{p+1}/(p+1)$ plays for integration: taking a unit finite difference recovers $x^p$.^[ref]

These two finite-sum transformations are inverses over any commutative ring. Substitute one sum into the other and use $\left(\genfrac{}{}{0ex}{}{n}{k}\right)\left(\genfrac{}{}{0ex}{}{k}{j}\right)=\left(\genfrac{}{}{0ex}{}{n}{j}\right)\left(\genfrac{}{}{0ex}{}{n-j}{k-j}\right)$. The inner alternating sum is $(1-1{)}^{n-j}$ by the binomial theorem, so only $j=n$ survives. In a Mahler expansion, $b_n$ is the nth finite difference of the sampled values $a_k$.

Brook Taylor was a mathematician whose 1715 work developed methods of finite differences and the expansion now called the Taylor series. He also studied vibrations of mechanical strings and mathematical perspective projection.^[ref] A Taylor polynomial uses the derivatives of a function at one point to build a local polynomial approximation. For instance, near zero, $e^x=1+x+x^2/2+O(x^3)$. The remainder is essential: a finite approximation can have a controlled error without asserting that an infinite Taylor series converges to the function. This distinction makes the method useful in numerical analysis and local analysis of differential equations.

A proposed modern measurement extends the weak acid dissociation measurement by estimating the local slope of an acid-base titration. Transfer a measured sample with a pipette into a stirred vessel; clamp a combination glass electrode/reference electrode clear of the stirring magnet and place a temperature probe nearby. A calibrated burette delivers a chemical base of independently standardized concentration $c_T$, such as sodium hydroxide. Record the initial sample volume $V_0$, cumulative delivered volume $v$, stable pH, and temperature after each small addition. Use a separate fresh portion for an acidward series. These are proposed measurement conditions, not reported experimental results.

For the instrument calibration, use fresh standard buffer solutions bracketing the measured pH range, with values appropriate to the actual temperature. The Hanna manual specifies separate rinse and calibration portions, gentle stirring, and approximately $4\mathrm{cm}$ immersion of the probe tip. It calls for stable readings; the sensing bulb and reference electrode junction must remain immersed. Rinse between standards and samples and avoid carrying rinse water into the measured portion.^[ref]

The readout pairs each added amount of substance with the measured pH. With negligible added volume, neighboring readings give the finite difference estimateassigned to the interval's midpoint. High buffer capacity appears as a small change of pH for a measured reagent dose. Choose increments small enough to resolve changing buffer capacity, but large enough that the change of pH exceeds measurement noise. Repeat with a fresh portion and different increment sizes; remeasure a standard afterward to check drift. A water blank can reveal contamination, but its pH must not be subtracted from the sample pH. These controls are proposed tests of the measurement and model assumptions.

If additions appreciably increase volume, even the analytical concentration changes: $C(v)=C_0{V}_0/(V_0+v)$. Merely dividing the added moles by the current volume does not restore the constant-volume derivative. One alternative is explicitly to report a changing-volume convention. For $c(v)=c_{T}v/(V_0+v)$, defineThis definition includes the reagent's changing concentration and differs from the constant-volume buffer capacity defined above.^[ref]

To compare with the constant-volume theory, fit the changing-volume chemical equilibrium model instead of treating the observed slope as $\beta$. In the same ideal monoprotic acid system, define $S_0=[{\mathrm{A}}^{-}{]}_0+[{\mathrm{OH}}^{-}{]}_0-[{\mathrm{H}}^{+}{]}_0$. For baseward additions, the electric charge balance at each reading isUse a minus sign before $c_{T}v$ for the corresponding acidward series. From the model parameters, evaluate the constant-volume buffer capacity at the chosen analytical concentration and pH. The fit also tests whether one acid dissociation constant describes the observations.

Limitations include incomplete mixing, slow glass electrode response, uncertain reagent concentration, and changes in temperature or ionic strength. Dissolved atmospheric carbon dioxide introduces an additional chemical acid system; comparable exposure and freshly prepared samples help distinguish it from the intended monoprotic acid response. Additional reacting chemical species and nonideal activity coefficients invalidate the simple formula. Especially when pH increments are tiny, measurement noise can dominate the inferred derivative; reporting the dose, pH interval, volume convention, and repeatability is part of reporting a meaningful buffer capacity.

Change-point detection estimates locations where a sequence's distribution or mean changes. A piecewise-constant mean has a sparse vector of adjacent differences; one-dimensional total variation denoising exploits this sparsity without fixing the number of changes in advance.

A circulant matrix applies the same rule at every position of a finite ring: moving both the input and output indices one step leaves its entries unchanged. Its first column determines the whole matrix. For a length-$N$ vector $c$, index rows and columns by $0,\dots ,N-1$ and defineThe indices use modular arithmetic. Thus each column is the previous column shifted down one place, with the last entry wrapping around to the top. Multiplication is exactly circular convolution:This makes a circulant matrix a natural model for a periodic finite difference rule or a filter acting on a periodic record. Strang's MIT lecture develops this connection between circulant matrices, circular convolution and the discrete Fourier transform.^[ref]

For example, the first column $c=(2,-1,0,-1{)}^T$ givesHere the indices wrap around modulo four. Each output compares a value with its two neighbors; this is the Graph Laplacian of a four-vertex cycle graph. A constant input gives zero, while an alternating input gives a large response. The Fourier diagonalization of a circulant matrix makes this distinction exact: its four eigenvalues are $0,2,4,2$.

The plotted discrete Fourier modes are complex vectors, shown through their real and imaginary parts. Each entry has modulus one; the Euclidean norm of each length-four vector is two. In particular, modes $k=1$ and $k=3$ have the same real part but opposite imaginary parts, so they are distinct eigenvectors with the same eigenvalue.

The useful surprise is that every length-$N$ circulant matrix shares the same basis of eigenvectors, whatever its first column. One change of basis therefore simplifies an entire family of linear maps, instead of requiring a separate eigenvalue computation for each one. Bamieh's tutorial develops the discrete Fourier transform from precisely this simultaneous diagonalization problem.^[ref]

Choose $N$ equal-amplitude, monochromatic odd components with orders $q_j=q_0+2j$, $j=0,\dots ,N-1$, and odd $q_0$. Fix the complex-field conventionHere $I$ is a normalized light intensity after averaging the fast carrier oscillations. Setting every ${\varphi }_j=0$ and summing a geometric series givesAt the removable singularities take the continuous limit. Maxima repeat every $T/2=\pi /\omega$; the first zeros on either side of a maximum lie at $t=\pm T/(2N)$. The zero-to-zero width is $T/N$, which is not the full width at half maximum.

Choose orders $13,15,17,19,21$ and driving wavelength $\lambda =800\mathrm{nm}$. Using $T=\lambda /c$ with the speed of light $c$ gives $T=2.66851\mathrm{fs}$ and burst spacing $1.33426\mathrm{fs}$. Solving the exact half-maximum equation gives a full width at half maximum of approximately $240.6$ attoseconds. These are values of the specified five-component model, not measurements from the 2001 experiment.

A deliberately chosen spectral phase ${\varphi }_q=q\omega t_0$ translates the whole attosecond pulse train by $t_0$ without changing its shape. Adding $\kappa (q-17{)}^2/2$ makes neighboring phase increments unequal: the same component intensities then produce a different temporal profile. For the choice $t_0=0.20\mathrm{fs}$ and $\kappa =0.08$ radians per squared harmonic order, the adjacent-component delay estimates arenamely $0.0981$, $0.1660$, $0.2340$ and $0.3019\mathrm{fs}$ for $q=13,15,17,19$. This finite difference estimates the group delay at the pair's midpoint. It is not a single common arrival time.

The infinite monochromatic sum is periodic forever and has infinite total energy. A real attosecond pulse train has a finite envelope and finite component linewidths. This calculation isolates coherent addition; it cannot describe an isolated burst or establish its contrast against satellites. As a further check, averaging $I(t)$ over one half-period gives $N|a{|}^2$ for any constant component spectral phases, because the cross terms integrate to zero by Fourier orthogonality. Spectral phase redistributes the light intensity in time while leaving that average fixed.

The common difference of an arithmetic progression is the fixed difference between consecutive sequence terms. It gives a finite difference that is independent of position. Checking one adjacent pair finds its value only when the arithmetic progression property is already known; otherwise, other pairs can disagree. For example, the differences in $7,10,14$ are three and four, so the three entries do not form an arithmetic progression.

Cyrus Samii is a political scientist at New York University whose methodological work concerns quantitative field research and causal inference. His academic biography connects the design of randomized controlled trials to empirical social research.^[ref]

With Peter M. Aronow, Samii studied sampling designs in which every unit can be observed but some pairs can never be observed together. The distinction matters: positive individual inclusion probabilities suffice for an unbiased estimator of a finite population total, whereas the usual Horvitz-Thompson variance estimator also needs pairwise inclusion positivity.^[ref]

An original two-unit example makes the single-draw variance identifiability obstruction explicit. Fix two unknown population values $y_1,y_2$ and choose one index $J$ with probabilities $1/2,1/2$. The Horvitz-Thompson estimator is $\hat{T}=2y_J$, soAn estimator based only on the selected index and its observed value has expected value of the form $[a(y_1)+b(y_2)]/2$. Such an additive expression cannot equal $(y_1-y_2{)}^2$ for every pair of real values. Taking the mixed finite difference at $(0,0),(1,0),(0,1),(1,1)$ gives zero for the additive expression butfor the desired variance. Thus a single draw permits an unbiased estimator of the total but no universally unbiased estimator of its sampling variance. Independent auxiliary randomization cannot repair this: averaging that randomness still gives functions $a$ and $b$ of the same observed data.

This is a restriction imposed by the observation mechanism, rather than a shortage of algebraic ingenuity. It motivates separating design-based inference about a total from the additional information needed to quantify its uncertainty.

Consider the original small Poisson equation exampleIts exact solution is the quadratic polynomial $u(x)=1+2x-x^2$. Use a uniform mesh $x_i=ih$ with $h=1/4$ and three interior unknowns $U_1,U_2,U_3$. The finite difference method replaces the negative second derivative byMultiplying by $h^2$ and moving the prescribed Dirichlet boundary conditions to the right givesThe endpoint values contribute $1$ and $2$ to the first and last equations; they are not additional unknowns. The Thomas algorithm givesand back substitution yieldsAll three values equal $u(x_i)$ exactly: the centered finite difference for the second derivative is exact on a quadratic polynomial. This special nodal exactness does not assert exactness for arbitrary forcing or for the piecewise-linear curve joining the nodes.

For any $n$ interior points with $h=1/(n+1)$, the scaled tridiagonal matrix has $b_i=2$ and $a_i=c_i=-1$. Its pivot elements are $p_i=(i+1)/i$: start at $p_1=2$ and use $p_i=2-1/p_{i-1}$. All are positive. Interior rows have equality in diagonal dominance, so strict diagonal dominance is not the reason this example works. Instead it is a symmetric positive-definite matrix, since for a real vector $v\ne 0$ with $v_0=v_{n+1}=0$,Zero would force every neighbouring value equal and hence all values zero. This connects the boundary constraints directly to the safe pivots for tridiagonal elimination.

For $-\Delta u=f$ on a square with homogeneous Dirichlet boundary conditions, the five-point finite difference operator on a mesh $h=1/N$ is diagonalized by $\sin (\pi ki/N)\sin (\pi \ell j/N)$. Its eigenvalues are ${\lambda }_{k\ell }=4h^{-2}[{\sin }^2(\pi k/(2N))+{\sin }^2(\pi \ell /(2N))]$. A two-dimensional discrete sine transform, division by these nonzero eigenvalues and the inverse transform give the solution. Odd extension realizes each one-dimensional transform by a Fast Fourier transform of length $2N$, making the total cost $O(N^2\log N)$ and storage $O(N^2)$. These finite-difference eigenvalues should not be confused with the continuous spectral eigenvalues ${\pi }^2(k^2+{\ell }^2)$.

A discrete Sobolev norm combines weighted norms of nodal values and their finite differences. For a real sequence $W_0,\dots ,W_N$ with zero endpoints, the displayed expression is a norm: it is the Euclidean norm of one vector containing both weighted lists, and zero norm forces all nodal values to vanish. It approximates the Sobolev space $H^1$ norm. Endpoint weights or prescribed zero endpoints must be specified when calling a discrete expression a norm.

In electrostatics, the electric potential $\Phi$ tells us the interaction potential energy per unit test electric charge in the electric field of prescribed sources. Its most directly useful quantity is an electric potential difference: how much work per unit electric charge an external agent must supply to move that test electric charge slowly between two points. This replaces the bookkeeping of a vector force along a path with the difference of two scalar values. The test electric charge must be small enough not to rearrange the sources appreciably.

For motion from $A$ to $B$, write $\Delta \Phi =\Phi (B)-\Phi (A)$. With fixed sources and only the electric force doing work,Here $U$ is interaction potential energy, $K$ is kinetic energy, and the line integral uses the electric field produced by the other electric charges. For slow transport with unchanged kinetic energy, the external work is $W_{\mathrm{ext}}=\Delta U$, so $\Delta \Phi =W_{\mathrm{ext}}/q$ for nonzero $q$. This external work has the opposite sign to the work done by the electric field. The SI unit is the volt,A volt measures energy per unit electric charge, not energy itself. For example, crossing an electric potential difference of $+5\mathrm{V}$ changes the potential energy of $q=+2\mathrm{nC}$ by $+10\mathrm{nJ}$, but changes that of $q=-2\mathrm{nC}$ by $-10\mathrm{nJ}$. Both test electric charges encounter the same electric potential difference; their opposite signs reverse the energy change.^[ref]

The electrostatic potential is a scalar function of position. The electric field is a conservative vector field, and all three components follow from the negative gradient of the electric potential:The electric field points toward decreasing electric potential. A positive test electric charge feels electric force in that direction; a negative test electric charge feels electric force in the opposite direction, since $\mathbf{F}=q\mathbf{E}$. A freely accelerated negative test electric charge therefore moves toward higher electric potential while lowering its potential energy $U=q\Phi$.^[ref]

Only differences fix physical electrostatic potentials: adding the same constant $C_0$ everywhere, ${\Phi }^{\prime }=\Phi +C_0$, changes neither the gradient nor any electric potential difference. Choosing an electrical reference potential is this elementary gauge transformation. Setting zero at infinity is convenient for finitely many localized point charges; it is not available for an ideal uniform electric field extending to infinity. There we can choose a finite reference point. Assigning zero at ground is a convention, whereas physically grounding an electrical conductor also allows exchange of electric charge to impose a boundary value. Zero electric potential at a point never by itself implies zero electric field.^[ref]

For a fixed point charge $Q$ at the origin in vacuum, choose zero at infinity. Coulomb's law giveswhere ${\epsilon }_0$ is the vacuum permittivity. Use the stipulated approximation $k=8.99\times 10^9\mathrm{N}{\mathrm{m}}^2/{\mathrm{C}}^2$ and $Q=+2\mathrm{nC}$. At $r=0.20\mathrm{m}$,A test electric charge $q=-3\mathrm{nC}$ has $U=-2.697\times 10^{-7}\mathrm{J}$ and radial electric force $F_r=-1.3485\times 10^{-6}\mathrm{N}$, directed inward. Moving it inward from $r_A=0.20\mathrm{m}$ to $r_B=0.10\mathrm{m}$ raises the electric potential from $89.9$ to $179.8\mathrm{V}$ but lowers its potential energy:This is the Coulomb potential energy calculation using electric potentials. The source's singular self-field at $r=0$ is excluded; when finding the force on a source point charge, use only the electric field of the other sources.^[ref]

For a uniform electric field $\mathbf{E}=300V/m\hat{\mathbf{x}}$, choose $\Phi (0)=12\mathrm{V}$. ThenMoving from $x_A=0$ to $x_B=0.04\mathrm{m}$ gives $\Delta \Phi =-12\mathrm{V}$. For $q=+2\mathrm{nC}$ the change in potential energy is $-24\mathrm{nJ}$ and the electric field does $+24\mathrm{nJ}$ of work. For $q=-2\mathrm{nC}$, the same prescribed displacement gives $\Delta U=+24\mathrm{nJ}$: the external agent must supply that work for slow transport. This uniform electric field approximates the interior of a parallel-plate capacitor sufficiently far from the edges; the fringing field of a capacitor limits the approximation.^[ref]

An equipotential is a level set $\Phi =\text{constant}$. On a smooth equipotential with nonzero gradient, every tangent displacement has $d\Phi =\nabla \Phi \cdot d\mathbf{l}=0$, so the electric field is normal to the equipotential. Thus moving a test electric charge along an equipotential requires no net electrical work. Around a point charge, the equipotentials are spheres; the circles in the diagram are their sections, not an assumption of two-dimensional Coulomb's law. A uniform electric field has parallel plane equipotentials. For contours separated by the same small electric potential difference, smaller normal separation means a stronger electric field, approximately $|\mathbf{E}|\approx |\Delta \Phi |/\Delta n$. At a point where $\nabla \Phi =0$, the electric field is zero and has no direction to call normal.^[ref]

The superposition principle adds electric potentials as signed scalars. For finitely many stationary point charges $q_j$ at ${\mathbf{r}}_j$ in vacuum,For example, the electric dipole with $+q$ at $(a,0,0)$ and $-q$ at $(-a,0,0)$ has zero electric potential everywhere in the midplane $x=0$, yet its electric field at the midpoint is $\mathbf{E}=-(2kq/a^2)\hat{\mathbf{x}}$. With $q=2\mathrm{nC}$ and $a=0.10\mathrm{m}$, this is $-3596N/C\hat{\mathbf{x}}$. At $(2a,0,0)$, the same model gives $\Phi =2kq/(3a)\approx 119.867\mathrm{V}$ and $E_x=8kq/(9a^2)\approx 1598.222N/C$. Cancellation of electric potentials is quite different from cancellation of vector electric fields.

A motivating measurement is conductive-paper electric potential mapping. A documented teaching experiment uses slightly conductive paper on an insulating board, two circular electrodes in highly conducting ink, a steady supply, and a digital voltmeter. Connect the supply to the two electrodes, keep the voltmeter common lead on the negative electrode, and use its other lead as a movable probe. Adjust the supply using the measured electric potential difference between electrodes; the published procedure uses $10\mathrm{V}$. The observable is the voltmeter reading $\Phi (P)-\Phi (\mathrm{COM})$ at the probe position $P$. Locate several points reading $2\mathrm{V}$, record their positions on a separate map, and join them into an equipotential; repeat at other readings. Lift the probe between contacts so it does not damage the paper. This procedure makes the electrical reference potential concrete: every reading compares two contacts.^[ref]

From neighboring readings infer an electric field component by a finite difference, $E_x\approx -[\Phi (x+h)-\Phi (x)]/h$. Reconstruct two in-plane components with displacements along two independent directions. Alternatively, draw the electric field normal to mapped equipotentials, toward lower electric potential. The second diagram illustrates the same measurement principle with bar electrodes spanning a uniform sheet, insulating top and bottom edges, and a synthetic model $\Phi (x)=(50V/m)x$. The model readings $4.0\mathrm{V}$ at $x=0.08\mathrm{m}$ and $5.0\mathrm{V}$ at $x=0.10\mathrm{m}$ imply $E_x=-50V/m$. These values are calculated illustrations, not reported experimental data. The documented circular-electrode experiment similarly estimates the electric field from readings at successive centimetre marks.^[ref]

Useful controls include checking the voltmeter zero, checking that the electric potential difference along each painted electrode is negligible, and repeating reference readings to detect supply drift. Reversing the supply should reverse the signs relative to the same reference contact. Contact resistance, leakage electric currents, and touching the sheet can distort the map. A voltmeter has finite input electrical resistance and draws a small electric current; voltmeter loading changes the quantity being measured if that electrical resistance is insufficiently large relative to the sheet's effective source electrical resistance.^[ref][ref]

The paper carries a steady electric current, so this is a planar analogue rather than a direct measurement of isolated point charges in three-dimensional vacuum. For constant electrical conductivity, Ohm's law and steady charge conservation give $\mathbf{j}=-\sigma \nabla \Phi$ and $\nabla \cdot \mathbf{j}=0$, hence the two-dimensional Laplace equation away from electrodes. Insulating sheet edges impose zero normal electric current, while finite electrodes fix boundary values. This derivation explains both the analogy and its limits: inhomogeneous paper, finite boundaries, and finite electrodes can change the contours. One should not fit the three-dimensional point charge formula $kQ/r$ to a planar map without testing that model. Small probe spacing resolves spatial variation better, but amplifies uncertainty in the inferred finite difference. The published laboratory instructions explicitly ask students to assess whether their proposed field fit is appropriate.^[ref][ref]

The scalar description has a crucial limit. Faraday's law of induction for a stationary closed path $\Gamma$ sayswhere ${\Phi }_B$ is magnetic flux. Nonzero changing magnetic flux gives nonzero circulation, incompatible with a single-valued electric potential alone. More generally, a scalar electric potential and a magnetic vector potential together giveConsequently, in the presence of induction, two voltmeter lead routes between the same contacts can produce different readings if the enclosed magnetic flux changes. This follows by subtracting the two path line integrals; it is a deduction from Faraday's law of induction, not a claimed outcome of the paper experiment. The steady setup and negligible induction justify interpreting its readings as electric potential differences. Merely introducing a scalar electric potential that depends on time cannot supply a nonzero curl of the electric field.^[ref]

A short mathematical justification connects the statements above. Along a smooth path, the chain rule gives $d\Phi =\nabla \Phi \cdot d\mathbf{l}$. Integrating and using $\mathbf{E}=-\nabla \Phi$ yields the endpoint formula by the fundamental theorem for line integrals, proving path independence. Conversely, vanishing circulation around every closed path lets us define $\Phi (P)=-{\int }_{P_0}^P\mathbf{E}\cdot d\mathbf{l}$ independently of the route, with $\Phi (P_0)=0$. Zero curl is a local test; for a continuously differentiable curl-free vector field, a simply connected open region suffices for a global potential. The explicit finite-source formula already gives a global electric potential away from its source points.^[ref]

On a uniform mesh $x_n=a+nh$, the centered second finite difference is exact for every quadratic polynomial. In particular, for $y^{\prime \prime }=c$ with endpoint boundary conditions, the unique discrete solution equals the exact quadratic solution at every mesh point. Subtracting consecutive first differences reduces the equation to $d_{n+1}-d_n=c{h}^2$, so $y_n=\frac{1}{2}c{h}^{2}n(n-1)+d_{1}n+y_0$. The second endpoint determines $d_1$. Uniqueness follows because any difference of two discrete solutions has zero second finite difference, hence is linear in the mesh index and vanishes at both endpoints. Exact nodal values in this example do not imply that a general finite difference method is exact for other ordinary differential equations.

An approximation to a Jacobian matrix obtained using finite differences instead of analytic partial derivatives. For a vector-valued function $F$, the forward-difference entry is $J_{ij}\approx (F_i(x+h{e}_j)-F_i(x))/h$, where $e_j$ is a coordinate basis vector. The step $h$ balances truncation and rounding errors; the resulting matrix can be used in a Newton root-finding iteration.

A finite difference operator maps grid values to weighted combinations of nearby grid values. Scaled finite differences approximate derivatives; their weights and grid spacing determine the order of a numerical method.

Adding one sequence term changes a partial sum by exactly that entry, so $S_{n+1}-S_n=a_{n+1}$. Taking another finite difference givesThus arithmetic sequence terms produce partial sums with constant second finite differences. Conversely, if these second finite differences are a fixed $d$ and $a_{n+1}=S_{n+1}-S_n$, then those entries form an arithmetic progression. For nonzero $d$, the displayed sum formula is a quadratic function; for $d=0$, it reduces to the linear expression $S_n=n{a}_1$.

The lowest positive normal mode frequency of an oscillating system is its fundamental frequency. For an ideal uniform string of length $L$ with fixed ends and phase speed $c$, $f_1=c/(2L)$. A spatial finite difference approximation has its own lowest positive normal mode frequency, approaching this continuous value as the grid is refined.