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The term "Maxwell's equations" is often also used for equivalent alternative formulations. Versions of Maxwell's equations based on the electric and magnetic scalar potentials are preferred for explicitly solving the equations as a boundary value problem, analytical mechanics, or for use in quantum mechanics.
[24] [25] Maxwell deals with the motion-related aspect of electromagnetic induction, v × B, in equation (77), which is the same as equation (D) in Maxwell's original equations as listed below. It is expressed today as the force law equation, F = q ( E + v × B ) , which sits adjacent to Maxwell's equations and bears the name Lorentz force ...
The structure of Maxwell relations is a statement of equality among the second derivatives for continuous functions. It follows directly from the fact that the order of differentiation of an analytic function of two variables is irrelevant (Schwarz theorem).
G = Gibbs free energy p = Pressure H = Enthalpy S = Entropy U = Internal energy V = Volume F = Helmholtz free energy T = Temperature. The thermodynamic square (also known as the thermodynamic wheel, Guggenheim scheme or Born square) is a mnemonic diagram attributed to Max Born and used to help determine thermodynamic relations.
The original equations used Hamilton's more expressive quaternion notation, [165] a kind of Clifford algebra, which fully subsumes the standard Maxwell vectorial equations largely used today. [166] In the late 1880s there was a debate over the relative merits of vector analysis and quaternions.
Electromagnetic behavior is governed by Maxwell's equations, and all parasitic extraction requires solving some form of Maxwell's equations. That form may be a simple analytic parallel plate capacitance equation or may involve a full numerical solution for a complex 3D geometry with wave propagation.
Some equations of physics are conformal invariant, e.g. the Maxwell's equations in source-free space, [6] but not all. The relevance of the conformal transformations in spacetime is not known at present, but the conformal group in two dimensions is highly relevant in conformal field theory and statistical mechanics. [7]
The first two integrals here are area I while the second two are the negative of area II. The two areas add to zero hence their magnitudes are equal according to this Gibbs criterion. This is again the equal area rule of Maxwell, the Maxwell construction, and it can also be shown analytically. Since () = +,