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Potential function

A potential function is a scalar-valued function \phi defined on a domain in Euclidean space such that its gradient \nabla \phi equals a given vector field \mathbf{F}, i.e., \mathbf{F} = \nabla \phi. This relationship characterizes \mathbf{F} as a conservative vector field, for which the line integral between two points is independent of the path taken and equals the difference in the potential function values at those points.^[1]^[2]^[3] In mathematics, particularly vector calculus, potential functions arise in the study of multivariable functions and enable the simplification of integrals over conservative fields, such as in Green's theorem and Stokes' theorem applications. The existence of a potential function for \mathbf{F} is equivalent to the vector field being irrotational, meaning its curl \nabla \times \mathbf{F} = \mathbf{0}, provided the domain is simply connected. Finding such a function involves integrating the components of \mathbf{F} and verifying consistency through partial derivatives, with the result determined up to an additive constant.^[4]^[5]^[6] In physics, the concept extends to conservative forces, where the force \mathbf{F} is typically expressed as the negative gradient of a potential energy function U, i.e., \mathbf{F} = -\nabla U, to reflect that work done by the force decreases the potential energy. This convention is central to fields like electrostatics, gravitation, and mechanics; for example, the gravitational potential energy near Earth's surface is U = mgh, where m is mass, g is gravitational acceleration, and h is height. Potential functions thus quantify stored energy in systems and facilitate the analysis of equilibrium and motion without tracking paths explicitly./02%3A_Electrostatic_Field_I/2.02%3A_The_Scalar_Potential_Function)^[7]^[8] The utility of potential functions spans beyond classical contexts, appearing in electromagnetism via scalar and vector potentials for solving Maxwell's equations, and in engineering for modeling fluid flow or stress fields under irrotational assumptions. Their defining property ensures path independence, making them indispensable for theoretical and computational predictions in conservative systems.^[9]^[10]

Mathematical Foundations

Scalar Potential

In vector calculus, a scalar potential is a scalar-valued function \phi defined on a domain such that a given vector field \mathbf{F} can be expressed as the gradient of \phi, i.e., \mathbf{F} = \nabla \phi.^[11] This mathematical convention is common in pure analysis, where \phi directly generates the field through differentiation. In physics, particularly for conservative forces, the convention often reverses the sign to reflect potential energy, yielding \mathbf{F} = -\nabla \phi, ensuring that the force points in the direction of decreasing potential.^[12] The existence of such a scalar potential is tied to the vector field being conservative, meaning the line integral \int_C \mathbf{F} \cdot d\mathbf{r} is independent of the path C connecting two points. This path independence holds if and only if the curl of \mathbf{F} vanishes, \nabla \times \mathbf{F} = \mathbf{0}, assuming the domain is simply connected and the components of \mathbf{F} are continuously differentiable. Under these conditions, the scalar potential can be constructed explicitly as \phi(\mathbf{x}) = -\int_{\mathbf{a}}^{\mathbf{x}} \mathbf{F} \cdot d\mathbf{r}, where the integral is taken along any path from a fixed reference point \mathbf{a} to \mathbf{x}; the negative sign aligns with the physics convention, though it is omitted in the mathematical one.^[11]^[12] The scalar potential is unique only up to an additive constant, as adding any constant c to \phi preserves the gradient: \nabla (\phi + c) = \nabla \phi. Fixing the value of \phi at a specific point, such as \phi(\mathbf{a}) = 0, removes this ambiguity and determines \phi uniquely throughout the domain.^[11] A representative example in two dimensions is the vector field \mathbf{F}(x, y) = \left( -\frac{y}{x^2 + y^2}, \frac{x}{x^2 + y^2} \right) defined for (x, y) \neq (0, 0). This field admits the scalar potential \phi(x, y) = \arctan(y/x), excluding branch cuts along the negative x-axis where the arctangent is discontinuous; verifying, \nabla \phi = \mathbf{F}. More broadly, the Helmholtz decomposition theorem asserts that any sufficiently smooth vector field in \mathbb{R}^3 (with suitable decay at infinity) can be uniquely decomposed as \mathbf{F} = -\nabla \phi + \nabla \times \mathbf{A}, separating it into an irrotational component -\nabla \phi (the negative gradient of a scalar potential) and a solenoidal component \nabla \times \mathbf{A} (the curl of a vector potential).^[13]

Vector Potential

In vector calculus, a vector potential \mathbf{A} for a vector field \mathbf{F} in three-dimensional space is defined as a vector field satisfying \mathbf{F} = \nabla \times \mathbf{A}, provided that \nabla \cdot \mathbf{F} = 0. This representation applies to solenoidal (divergence-free) fields and contrasts with the scalar potential used for irrotational (curl-free) fields. The condition \nabla \cdot \mathbf{F} = 0 ensures the existence of \mathbf{A} in simply connected domains, as guaranteed by fundamental theorems of vector analysis.^[14]^[15] The existence of the vector potential follows from Stokes' theorem, which relates the surface integral of \mathbf{F} over any surface S bounded by a closed curve C to the line integral of \mathbf{A} around C: \oint_C \mathbf{A} \cdot d\mathbf{l} = \iint_S \mathbf{F} \cdot d\mathbf{S}. For \nabla \cdot \mathbf{F} = 0, the flux of \mathbf{F} through any closed surface vanishes by the divergence theorem, implying consistency in defining \mathbf{A} such that the circulation of \mathbf{F} is path-independent under suitable boundary conditions, like those in a simply connected region. This allows explicit construction of \mathbf{A} via integrals over surfaces spanning the relevant curves.^[16] The vector potential \mathbf{A} is not unique due to gauge freedom: if \mathbf{A} satisfies \nabla \times \mathbf{A} = \mathbf{F}, then \mathbf{A}' = \mathbf{A} + \nabla \chi for any scalar function \chi also works, since \nabla \times (\nabla \chi) = \mathbf{0}. To specify a unique \mathbf{A}, gauge conditions are imposed, such as the Coulomb gauge \nabla \cdot \mathbf{A} = 0, which simplifies computations by making \mathbf{A} divergence-free, or the Lorenz gauge \nabla \cdot \mathbf{A} = 0 in static cases (with extensions for dynamics). These choices preserve the physical field \mathbf{F} while aiding solvability of the underlying equations.^[17] A concrete example in three dimensions is a uniform solenoidal field \mathbf{B} = (0, 0, B), for which \mathbf{A} = \left( -\frac{B y}{2}, \frac{B x}{2}, 0 \right) satisfies \nabla \times \mathbf{A} = \mathbf{B}, as direct computation yields the components (\partial A_z / \partial y - \partial A_y / \partial z, \partial A_x / \partial z - \partial A_z / \partial x, \partial A_y / \partial x - \partial A_x / \partial y) = (0, 0, B). This form illustrates the rotational symmetry inherent in such potentials.^[18] The vector potential plays a key role in the Helmholtz decomposition theorem, which uniquely (up to boundary conditions) expresses any sufficiently smooth vector field \mathbf{F} as \mathbf{F} = -\nabla \phi + \nabla \times \mathbf{A}, where -\nabla \phi is the irrotational component and \nabla \times \mathbf{A} is the solenoidal component with \nabla \cdot \mathbf{A} = 0. This decomposition separates the field's curl-free and divergence-free parts, with the latter always representable via a vector potential.^[19]^[13]

Physical Interpretations

Conservative Force Fields

In physics, a force field \mathbf{F} is defined as conservative if the work done by the force along any closed path C is zero, expressed mathematically as \oint_C \mathbf{F} \cdot d\mathbf{r} = 0.^[20] This path independence implies that the work depends only on the initial and final positions, not the trajectory taken between them.^[21] A key consequence is that such fields can be derived from a scalar potential function \phi, where \mathbf{F} = -\nabla \phi, with \phi typically representing potential energy per unit mass or charge.^[22] This equivalence holds in three-dimensional Euclidean space for continuously differentiable fields, linking the conservative nature directly to the existence of the potential.^[23] The energy interpretation of conservative fields underscores their role in mechanics: the change in potential energy \Delta \phi between two points equals the negative of the work W done by the field, \Delta \phi = -W.^[24] In an isolated system where only conservative forces act, this relation ensures conservation of total mechanical energy, as the decrease in potential energy converts directly to kinetic energy without dissipation.^[20] For instance, in gravitational or elastic systems, this conservation principle simplifies analysis by allowing energy balances rather than tracking forces along paths.^[25] When conservative fields arise from distributed sources, such as charges or masses, the potential \phi satisfies Poisson's equation, a generalization that incorporates source density; for example, in electrostatics, \nabla^2 \phi = -\rho / \epsilon_0, where \rho is charge density and \epsilon_0 is the permittivity of free space.^[26] This equation maintains the conservative property by ensuring the field remains the negative gradient of \phi, even in the presence of sources, while regions without sources reduce to Laplace's equation \nabla^2 \phi = 0.^[27] The focus here remains on how this framework upholds path-independent work and energy conservation, rather than specific solutions. The use of potential functions in conservative fields originated in analytical mechanics, introduced by Joseph-Louis Lagrange in the late 18th century through his formulation of dynamics in terms of generalized coordinates and energies, and further developed by William Rowan Hamilton in the early 19th century with his principle of least action.^[28] These approaches replaced direct force calculations with variational principles involving potentials, enabling elegant treatments of complex systems like planetary motion.^[29] A classic example is the one-dimensional harmonic oscillator, such as a mass-spring system, where the restoring force is \mathbf{F} = -k x \hat{i} and the corresponding potential energy function is \phi = \frac{1}{2} k x^2.^[20] Here, the force derives directly as F_x = -\frac{d\phi}{dx} = -k x, illustrating how the potential encapsulates the field's conservative behavior and enables energy conservation analyses.^[25] This irrotational condition (\nabla \times \mathbf{F} = 0), prerequisite from vector calculus foundations, aligns with the conservative definition but emphasizes the physical energy implications in this context.^[30]

Irrotational and Solenoidal Fields

Note that sign conventions for potentials vary by context: in mathematical vector analysis, the scalar potential \phi is often defined such that an irrotational vector field \mathbf{F} satisfies \mathbf{F} = \nabla \phi, while in physics for conservative force fields, the convention \mathbf{F} = -\nabla \phi is standard to ensure the work done by the field equals the decrease in potential energy. Other physical fields, like velocity in irrotational fluid flow, may use the positive gradient. This difference arises from the perspective of energy storage in physics versus general path independence in mathematics.^[31] In vector analysis, an irrotational vector field \mathbf{F} satisfies \nabla \times \mathbf{F} = \mathbf{0} everywhere in its domain, implying that the field is locally conservative, meaning the line integral of \mathbf{F} around any infinitesimal closed loop vanishes.^[32] This condition ensures that \mathbf{F} can be expressed as the gradient of a scalar potential \phi, i.e., \mathbf{F} = \nabla \phi, at least locally. A fundamental vector calculus identity underpinning this is \nabla \times (\nabla \phi) = \mathbf{0} for any sufficiently smooth scalar \phi, which holds due to the equality of mixed partial derivatives and the antisymmetric nature of the curl operator.^[33] For global existence of the scalar potential in \mathbb{R}^3, the domain must be simply connected, such as a star-shaped region or the entire space minus no obstacles; in such domains, if \nabla \times \mathbf{F} = \mathbf{0} and \mathbf{F} is continuously differentiable, then \mathbf{F} = \nabla \phi holds throughout.^[34] In multiply connected domains, like \mathbb{R}^3 excluding a torus or a solid obstacle, the potential may fail to exist globally unless additional conditions are met, specifically that the circulation \oint_C \mathbf{F} \cdot d\mathbf{l} = 0 for every closed curve C that is not contractible to a point within the domain (e.g., loops encircling "holes").^[35] Violation of this flux-like condition through non-trivial cycles leads to multi-valued potentials or none at all. Dually, a solenoidal vector field \mathbf{F} satisfies \nabla \cdot \mathbf{F} = \mathbf{0}, indicating no sources or sinks, and thus can be expressed locally as the curl of a vector potential \mathbf{A}, i.e., \mathbf{F} = \nabla \times \mathbf{A}.^[33] The justifying identity is \nabla \cdot (\nabla \times \mathbf{A}) = 0 for any smooth \mathbf{A}, arising from the divergence acting on the antisymmetric curl components, which cancel out. In domains without holes—analogous to simply connected but for the dual topology, such as contractible regions in \mathbb{R}^3—global existence of \mathbf{A} is guaranteed if \mathbf{F} is smooth and solenoidal. For multiply connected regions, global \mathbf{A} requires that the flux \iint_S \mathbf{F} \cdot d\mathbf{S} = 0 through any closed surface S bounding a non-trivial hole, ensuring compatibility with the topology.^[33] Modern generalizations extend these results to smooth manifolds beyond Euclidean space, where the existence of potentials is governed by de Rham cohomology: a closed k-form \omega (analogous to \mathbf{F} via the metric) is exact (i.e., \omega = d\eta for some (k-1)-form \eta, corresponding to a potential) if and only if its cohomology class [\omega] \in H^k_{dR}(M) is zero, with non-trivial classes measuring topological obstructions like holes.^[36] This framework, developed in the 1930s–1940s, unifies the simply connected case (where H^1(M) = 0) with multiply connected ones via the dimensions of cohomology groups, providing a precise algebraic tool for potential existence on arbitrary manifolds.^[37]

Applications in Physics

Electrostatics and Magnetostatics

In electrostatics, the electric field \vec{E} is conservative and irrotational, allowing it to be expressed as the negative gradient of a scalar potential V, such that \vec{E} = -\nabla V.^[38] This relation follows from Maxwell's equations in the static limit, where \nabla \times \vec{E} = 0. The scalar potential V satisfies Poisson's equation \nabla^2 V = -\rho / \epsilon_0 in regions with charge density \rho, but in charge-free regions, it reduces to Laplace's equation \nabla^2 V = 0./05%3A_Electrostatics/5.15%3A_Poissons_and_Laplaces_Equations) For a single point charge q at the origin, the potential at a distance r is given by V = \frac{q}{4\pi \epsilon_0 r}, which is derived by integrating the electric field along a radial path from infinity.^[39] Boundary value problems in electrostatics involve solving Laplace's equation subject to specified conditions on a bounding surface. The uniqueness theorem guarantees that the solution is unique for Dirichlet boundary conditions, where V is specified on the surface, or for Neumann boundary conditions, where the normal derivative \partial V / \partial n (proportional to the surface charge) is specified. This theorem is proven using Green's first identity, showing that any two solutions differing by a function \psi must satisfy \int (\nabla \psi)^2 d\tau = 0, implying \psi = 0. Modern formulations use SI units, where \epsilon_0 is the vacuum permittivity (approximately $8.85 \times 10^{-12} \, \mathrm{F/m}), marking a shift from the 19th-century CGS system—adopted by early electromagnetism researchers—to the internationally standardized SI system in the mid-20th century for greater coherence in engineering and physics applications.^[40]^[41] As an example, consider the potential due to an electric dipole consisting of charges +q and -q separated by a small displacement \vec{d}, with dipole moment \vec{p} = q \vec{d}. At large distances r \gg d, the scalar potential approximates V \approx \frac{\vec{p} \cdot \hat{r}}{4\pi \epsilon_0 r^2}, obtained via multipole expansion of the point-charge formula.^[42] In magnetostatics, the magnetic field \vec{B} has zero divergence, \nabla \cdot \vec{B} = 0, allowing it to be represented as the curl of a vector potential \vec{A}, such that \vec{B} = \nabla \times \vec{A}.^[18] This decomposition satisfies Maxwell's static equation \nabla \cdot \vec{B} = 0 identically, since the divergence of a curl vanishes. From Ampère's law in the static case, \nabla \times \vec{B} = \mu_0 \vec{J}, substituting the expression for \vec{B} yields a equation for \vec{A}. In the Coulomb gauge, defined by \nabla \cdot \vec{A} = 0, the vector potential satisfies the vector Poisson equation \nabla^2 \vec{A} = -\mu_0 \vec{J} in regions with current density \vec{J}, or Laplace's equation \nabla^2 \vec{A} = 0 in current-free regions.^[43] Here, \mu_0 is the vacuum permeability (approximately $4\pi \times 10^{-7} \, \mathrm{H/m}) in SI units, consistent with the 20th-century transition from CGS conventions.^[40] The choice of Coulomb gauge simplifies calculations, as it decouples the components of \vec{A} in Cartesian coordinates and aligns with the instantaneous nature of static fields.

Gravitational Potential

In Newtonian gravity, the gravitational field \mathbf{g} is expressed as the negative gradient of a scalar potential \Phi, such that \mathbf{g} = -\nabla \Phi. For a point mass M, the potential takes the form \Phi = -\frac{[G](/page/G)M}{r}, where G is the gravitational constant and r is the distance from the mass.^[44] This potential satisfies Poisson's equation \nabla^2 \Phi = 4\pi G \rho, where \rho is the mass density, linking the curvature of the potential to the distribution of mass. Equipotential surfaces are the level sets of \Phi, where the gravitational potential energy per unit mass remains constant. On these surfaces, the gravitational force is perpendicular to the surface, and no work is done against gravity when moving a test mass along them.^[45] In orbital mechanics, the gravitational potential contributes to the effective potential \Phi_\text{eff} = \Phi + \frac{L^2}{2mr^2}, where L is the angular momentum and m is the reduced mass, enabling the derivation of Kepler's laws for bound orbits under inverse-square forces.^[46]^[47] The weak field approximation extends this to general relativity, where the metric component g_{00} \approx 1 + \frac{2\Phi}{c^2} for slow motions and weak fields, with c the speed of light.^[48] This form accounts for relativistic effects like the Shapiro time delay, first predicted in 1964 as an extra propagation delay for signals passing near a massive body. For Earth, the gravitational potential is approximately \Phi \approx -\frac{GM}{r}, and the geoid represents an equipotential surface closely matching mean sea level, undulating due to mass irregularities.^[49]

Uses in Other Disciplines

Optimization Algorithms

In amortized analysis of algorithms, particularly for data structures, a potential function \Phi is defined to account for latent costs or "prepaid work" that offsets expensive operations over a sequence of steps. The amortized cost of the i-th operation is given by \hat{c}_i = c_i + \Phi(D_i) - \Phi(D_{i-1}), where c_i is the actual cost and D_i is the data structure's state after the operation; this ensures that the total cost over n operations is \sum_{i=1}^n c_i + \Phi(D_n) - \Phi(D_0), bounding the average cost per operation when \Phi is chosen such that \Phi(D_i) \geq 0 and initial/final potentials are controlled.^[50] This method guarantees average O(1) time for operations like insertions in dynamic arrays, where resizing (copying elements to a larger array) is infrequent but costly.^[50] A classic example is the dynamic array (or resizable array) implementation, such as in stack operations, where the array doubles in capacity when full. Define the potential \Phi = 2 \times (number of elements) - current capacity, so that after an insertion without resize, \hat{c}_i = 1 + 2 = 3 (actual cost 1 plus potential increase of 2, as elements increase by 1), while during resize (actual cost O(m) for capacity m), the potential drops by approximately m (as capacity doubles from m to $2m while elements increase from m to m+1), yielding \hat{c}_i \approx 3. Over a sequence filling the array, the telescoping sum of potentials spreads the resize cost, achieving amortized O(1) per insertion despite occasional O(n) spikes. This approach, introduced by Sleator and Tarjan in 1985 for analyzing list update and paging algorithms, formalized the potential method building on earlier amortized ideas in data structures like splay trees.^[50] In continuous optimization, particularly gradient descent, the objective function f serves analogously as a potential to minimize, with updates x_{k+1} = x_k - \alpha \nabla f(x_k) descending the "potential hill" toward lower values; for convex f with Lipschitz gradient, this converges to the global minimum at rate O(1/k) under appropriate step size \alpha.^[51] The method traces to Cauchy's 1847 work on steepest descent but gained prominence in machine learning post-2010 for training models via empirical risk minimization. In AI applications, potential functions extend to non-convex losses (e.g., deep neural networks), where auxiliary potentials like f(x) + \frac{\|x\|^2}{2} regularize the landscape, enabling convergence proofs to stationary points via stochastic gradient descent despite local minima and saddles; regularization terms, such as L2 penalties, further smooth non-convexity to promote generalization in training large models.^[51]

Potential Games

In game theory, a potential game is a type of strategic-form game characterized by the existence of a scalar potential function P: S \to \mathbb{R}, where S = \prod_{i=1}^n S_i is the joint strategy space for n players with individual strategy sets S_i, such that unilateral deviations by any player i induce identical changes in their utility and the potential. Formally, P is an exact potential function if, for all players i, strategies s_i, s_i' \in S_i, and fixed strategies of others s_{-i} \in \prod_{j \neq i} S_j,

u_i(s_i', s_{-i}) - u_i(s_i, s_{-i}) = P(s_i', s_{-i}) - P(s_i, s_{-i}),

where u_i denotes player i's utility function. This equivalence ensures that the potential aggregates and reflects the incentive structure of all players' payoff changes from strategy deviations. The concept was introduced by Monderer and Shapley in their seminal 1996 paper, which formalized potential games as a class admitting such functions and established foundational properties for equilibrium analysis.^[52] A related but weaker variant is the ordinal potential game, where the potential need only preserve the ordinal ranking of deviations rather than their exact magnitudes. Here, the sign of the utility change matches the sign of the potential change:

\operatorname{sign}\left(u_i(s_i', s_{-i}) - u_i(s_i, s_{-i})\right) = \operatorname{sign}\left(P(s_i', s_{-i}) - P(s_i, s_{-i})\right),

for all i, s_i, s_i', s_{-i}, with the understanding that zero changes align accordingly. This notion, developed by Ui in 1999, broadens the applicability of potential-based analysis to games where precise payoff differences are unavailable or unnecessary, focusing instead on the direction of incentives. Ordinal potentials suffice for many convergence results while relaxing the stringency of exact matching.^[53] In exact potential games, pure-strategy Nash equilibria coincide with the local maxima of the potential function P, as any profitable unilateral deviation by a player would increase P, contradicting maximality. This mapping simplifies equilibrium detection, reducing the problem to optimizing a single scalar function. Furthermore, for finite exact potential games, best-response dynamics—where players sequentially select best responses to others' strategies—converge to a pure Nash equilibrium, a property proven by Monderer and Shapley that guarantees finite termination and stability under myopic play. Similar convergence holds for ordinal potential games under best-response dynamics, though potentially to approximate equilibria.^[52] A canonical example of an exact potential game is the congestion game, originally defined by Rosenthal in 1973, where players select subsets of resources (e.g., routes in a network) subject to increasing costs based on usage levels. In such games, the potential function is P(s) = \sum_{r \in R} \sum_{k=1}^{n_r(s)} c_r(k), with R the set of resources, n_r(s) the number of players using resource r under strategy profile s, and c_r(k) the cost of r when used by k players; this equals the total sum of players' costs, ensuring \Delta u_i = \Delta P for deviations. Congestion games model applications like traffic routing, where drivers choose paths to minimize travel time amid congestion, and the potential facilitates proving equilibrium existence and efficiency bounds. Monderer and Shapley later showed that finite congestion games are precisely the finite exact potential games with finite improvements property.^[52] Following the 1996 introduction, applications of potential games expanded significantly post-2000 in economics and network theory, including analyses of network formation, resource allocation in wireless systems, and coordination in social networks, leveraging the potential for tractable equilibrium computation and dynamics.^[54]

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Abstract. The de Rham cohomology is a cohomology based on differential forms on a smooth manifold. It uses the exterior derivative as the boundary.Missing: source | Show results with:source
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[PDF] two proofs of the de rham theorem - University of Notre Dame
The main goal of this paper is to state and prove the De Rham Theorem in two different ways. We will work exclusively in the realm of smooth manifolds, and we ...<|control11|><|separator|>
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[PDF] 8.022 (E&M) – Lecture 3
grad φ φ (x,y). → E=-gradφ points downhill. (x,y)=sin(x)sin(y), calculate its gradient. cos( sin cos. y x. The gradient always points uphill. Same potential φ ...
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[PDF] PHY481 - Lecture 7: The electrostatic potential and potential energy
(i) Electric potential due to a point charge. The electric field due to a point charge is. ~E = kQr r2. (10). The electric potential due to a point charge is ...
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Units for Magnetic Quantities | NIST
Sep 28, 2021 · The centimeter-gram-second (CGS) system of units was adopted by the pioneers of electromagnetism in the nineteenth century.
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History of the SI - IEC - International Electrotechnical Commission
After adopting the CGS system, the same commission also decided, in 1874, to adopt the ohm as the unit for resistance and the volt for electromotive force ( ...Missing: shift | Show results with:shift
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[PDF] EM 3 Section 5: Electric Dipoles
An electric dipole is formed by two point charges +q and −q connected by a vector a. The electric dipole moment is defined as p = qa .
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[PDF] 3. Magnetostatics
Here A is called the vector potential. While magnetic fields that can be written in the form (3.9) certainly satisfy r · B = 0, the converse is also true; any ...
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The Newtonian potential and the demagnetizing factors of ... - Journals
Jun 1, 2016 · The aim of this paper is to give a modern and concise derivation for the expression of the interior and exterior Newtonian potential (induced by ...
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[PDF] Gravity 3 - Gravitational Potential and the Geoid
Since potential energy, U = gh on an equipotential surface, and U is constant by definition, any change in gravity corresponds to a change in height, h. This ...
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[PDF] Effective Potential - UCSB Physics
kinetic energy involving φ into the potential, creating an effective potential, since it is only a function of r, namely −1. 2 mr2 ˙ φ2 = − L2. 2mr2 (the ...
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[PDF] 3.4 Kepler's laws - Princeton University
There is a minimum effective po- tential energy for radial motion, as can be seen in Fig 3.1, If the total energy is equal to this minimum, then there can be no ...Missing: Φ + | Show results with:Φ +
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[PDF] First Order Approximations in General Relativity
Dec 13, 2018 · τb = τa. 1 -. Φ(xb) - Φ(xa) c2. (11). Therefore, when we derive our weak field metric, it must be consistent with this expression. ... We can then ...
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[PDF] Geoid Surfaces and Theory - National Geodetic Survey
A geoid is the equipotential surface of Earth's gravity field best fitting mean sea level, undulating about the ellipsoid, and cannot be seen directly.
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[PDF] Amortized Efficiency Of List Update and Paging Rules
with potential a', we define the amortized time of the operation to be t + CP' - CP. That is, the amortized time of an operation is its running time plus ...
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[1712.04581] Potential-Function Proofs for First-Order Methods - arXiv
Dec 13, 2017 · This note discusses proofs for convergence of first-order methods based on simple potential-function arguments. We cover methods like gradient ...
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Potential Games - ScienceDirect
We define and discuss several notions of potential functions for games in strategic form. We characterize games that have a potential function, and we present ...
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Potential games: a purely ordinal approach - ScienceDirect.com
As a slight revision of Monderer and Shapley's definition, a potential for a strategic game is defined as a strict order on the set of strategy profiles, ...
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[PDF] Network Formation Games and the Potential Function Method
May 23, 2007 · Such connection games have been extensively studied in the economics literature (see Jackson (2006) for a survey) to model social network ...