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Jefimenko says, "...neither Maxwell's equations nor their solutions indicate an existence of causal links between electric and magnetic fields. Therefore, we must conclude that an electromagnetic field is a dual entity always having an electric and a magnetic component simultaneously created by their common sources: time-variable electric ...
The convection–diffusion equation describes the flow of heat, particles, or other physical quantities in situations where there is both diffusion and convection or advection. For information about the equation, its derivation, and its conceptual importance and consequences, see the main article convection–diffusion equation. This article ...
Even after such symmetry reductions, the reduced system of equations is often difficult to solve. For example, the Ernst equation is a nonlinear partial differential equation somewhat resembling the nonlinear Schrödinger equation (NLS). But recall that the conformal group on Minkowski spacetime is the symmetry group of the Maxwell equations.
Physically, hydrostatic equilibrium connects the gradient of the potential, the density, and the gradient of the pressure, whereas Poisson's equation connects the potential with the density. Thus, if we have a further equation that dictates how the pressure and density vary with respect to one another, we can reach a solution.
The Higgs field φ satisfies the Klein–Gordon equation. The weak interaction fields Z, W ± satisfy the Proca equation. These equations can be solved exactly. One usually does so by considering first solutions that are periodic with some period L along each spatial axis; later taking the limit: L → ∞ will lift this periodicity restriction.
The porous medium equation name originates from its use in describing the flow of an ideal gas in a homogeneous porous medium. [6] We require three equations to completely specify the medium's density , flow velocity field , and pressure : the continuity equation for conservation of mass; Darcy's law for flow in a porous medium; and the ideal gas equation of state.
Combining the Vlasov equation with the Poisson equation for gravity: =. gives the Jeans equations. More explicitly, If n = n ( x , t ) is the density of stars in space, as a function of position x = ( x 1 , x 2 , x 3 ) and time t , v = ( v 1 , v 2 , v 3 ) is the velocity, and Φ = Φ( x , t ) is the gravitational potential , the Jeans equations ...
The equation is derived by solving the Einstein equations for a general time-invariant, spherically symmetric metric. For a solution to the Tolman–Oppenheimer–Volkoff equation, this metric will take the form [1]