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The source free equations can be written by the action of the exterior derivative on this 2-form. But for the equations with source terms (Gauss's law and the Ampère-Maxwell equation), the Hodge dual of this 2-form is needed. The Hodge star operator takes a p-form to a (n − p)-form, where n is the number of dimensions.
A flux in classical mechanics is normally not a governing equation, but usually a defining equation for transport properties. Darcy's law was originally established as an empirical equation, but is later shown to be derivable as an approximation of Navier-Stokes equation combined with an empirical composite friction force term. This explains ...
An example of mathematical physics: solutions of Schrödinger's equation for quantum harmonic oscillators (left) with their amplitudes (right).. Mathematical physics refers to the development of mathematical methods for application to problems in physics.
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
In electromagnetism, Jefimenko's equations (named after Oleg D. Jefimenko) give the electric field and magnetic field due to a distribution of electric charges and electric current in space, that takes into account the propagation delay (retarded time) of the fields due to the finite speed of light and relativistic effects.
The general form of wavefunction for a system of particles, each with position r i and z-component of spin s z i. Sums are over the discrete variable s z , integrals over continuous positions r . For clarity and brevity, the coordinates are collected into tuples, the indices label the particles (which cannot be done physically, but is ...
Specifically, the more general form of the Hamilton's equation reads = {,} +, where f is some function of p and q, and H is the Hamiltonian. To find out the rules for evaluating a Poisson bracket without resorting to differential equations, see Lie algebra ; a Poisson bracket is the name for the Lie bracket in a Poisson algebra .
In numerical analysis, leapfrog integration is a method for numerically integrating differential equations of the form ¨ = = (), or equivalently of the form ˙ = = (), ˙ = =, particularly in the case of a dynamical system of classical mechanics.