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The scalar potential is an example of a scalar field. Given a vector field F, the scalar potential P is defined such that: [1] = = (,,), where ∇P is the gradient of P and the second part of the equation is minus the gradient for a function of the Cartesian coordinates x, y, z. [a] In some cases, mathematicians may use a positive sign in front ...
A particular choice of the scalar and vector potentials is a gauge (more precisely, gauge potential) and a scalar function ψ used to change the gauge is called a gauge function. [citation needed] The existence of arbitrary numbers of gauge functions ψ(r, t) corresponds to the U(1) gauge freedom of this theory. Gauge fixing can be done in many ...
Introducing the electric potential φ (a scalar potential) and the magnetic potential A (a vector potential) defined from the E and B fields by: =, =.. The four Maxwell's equations in a vacuum with charge ρ and current J sources reduce to two equations, Gauss's law for electricity is: + =, where here is the Laplacian applied on scalar functions, and the Ampère-Maxwell law is: (+) = where ...
The Liénard–Wiechert potentials describe the classical electromagnetic effect of a moving electric point charge in terms of a vector potential and a scalar potential in the Lorenz gauge. Stemming directly from Maxwell's equations , these describe the complete, relativistically correct, time-varying electromagnetic field for a point charge in ...
An action of such a gravitational scalar–tensor theory can be written as follows: = [() () + (,)], where is the metric determinant, is the Ricci scalar constructed from the metric , is a coupling constant with the dimensions , () is the scalar-field potential, is the material Lagrangian and represents the non-gravitational fields.
By placing φ as potential, ∇φ is a conservative field. Work done by conservative forces does not depend on the path followed by the object, but only the end points, as the above equation shows. The gradient theorem also has an interesting converse: any path-independent vector field can be expressed as the gradient of a scalar field. Just ...
A mathematical function, whose values are given by a scalar potential or vector potential; The electric potential, in the context of electrodynamics, is formally described by both a scalar electrostatic potential and a magnetic vector potential; The class of functions known as harmonic functions, which are the topic of study in potential theory
Mathematically, a scalar field on a region U is a real or complex-valued function or distribution on U. [1] [2] The region U may be a set in some Euclidean space, Minkowski space, or more generally a subset of a manifold, and it is typical in mathematics to impose further conditions on the field, such that it be continuous or often continuously differentiable to some order.