For example, the analysis of radio antennas makes full use of Maxwell's vector and scalar potentials to separate the variables, a common technique used in formulating the solutions of differential equations.
32.
The term " potential theory " arises from the fact that, in 19th century physics, the fundamental forces of nature were believed to be derived from scalar potentials which satisfied Laplace's equation.
33.
Using the Lorenz gauge, Maxwell's equations can be written compactly in terms of the magnetic vector potential "'A "'and the electric scalar potential " ? ":
34.
The most common generalization of the linear theory above is to add a scalar potential to the Lagrangian, where typically, in addition to a mass term, " V " is a polynomial in.
35.
Because the force field is conservative, there is a scalar potential energy per unit mass, " ? ", at each point in space associated with the force fields; this is called gravitational potential.
36.
Applying an electric field " F " to the superlattice structure causes the Hamiltonian to exhibit an additional scalar potential " e? ( z ) = " eFz " that destroys the translational invariance.
37.
In such cases the way in which the different supersymmetry content enters into the moduli behaviour is through the flux dependent scalar potential the moduli experience, the N = 1 case is different from the N = 2 case.
38.
The final line shows that if the two scalar potentials differ at the initial time by more than a spatially independent function, then the current densities that the potentials generate will differ infinitesimally after " t " 0.
39.
The gradient of the scalar potential ( and hence also its opposite, as in the case of a vector field with an associated potential field ) is everywhere perpendicular to the equipotential surface, and zero inside a three-dimensional equipotential region.
40.
For a charged particle " q " in an electromagnetic field, described by the scalar potential " ? " and vector potential "'A "', there are two parts to the Hamiltonian to substitute for.
