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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).
The fundamental thermodynamic relation describing the system will then be of the form = (,,,). In the more general case where the paramagnet does not share an axis with the magnetic field, the extensive parameters characterizing the magnetic state will be I x , I y , I z {\displaystyle I_{x},I_{y},I_{z}} .
Here, the 3-form J is called the electric current form or current 3-form: =. That F is a closed form , and the exterior derivative of its Hodge dual is the current 3-form, express Maxwell's equations: [ 4 ]
The thermodynamic square can also be used to find the first-order derivatives in the common Maxwell relations.The following procedure should be considered: Looking at the four corners of the square and make a shape with the quantities of interest.
In the differential form formulation on arbitrary space times, F = 1 / 2 F αβ dx α ∧ dx β is the electromagnetic tensor considered as a 2-form, A = A α dx α is the potential 1-form, = is the current 3-form, d is the exterior derivative, and is the Hodge star on forms defined (up to its orientation, i.e. its sign) by the ...
The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations and the Lorentz force) in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems. These ...