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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.
Continuous charge distribution. The volume charge density ρ is the amount of charge per unit volume (cube), surface charge density σ is amount per unit surface area (circle) with outward unit normal nĚ‚, d is the dipole moment between two point charges, the volume density of these is the polarization density P.
Integral equation methods, however, generate dense (all entries are nonzero) linear systems, making such methods preferable to FD or FEM only for small problems. Such systems require O(n 2) memory to store and O(n 3) to solve via direct Gaussian elimination or, at best, O(n 2) if solved iteratively. Increasing circuit speeds and densities ...
Momentum is 3-D planar EM simulation software [1] for electronics and antenna analysis, a partial differential equation solver of Maxwell's equations based on the method of moments. [2] It is a 3-D planar electromagnetic (EM) simulator used for passive circuit analysis.
Maxwell's equations can directly give inhomogeneous wave equations for the electric field E and magnetic field B. [1] Substituting Gauss's law for electricity and Ampère's law into the curl of Faraday's law of induction, and using the curl of the curl identity ∇ × (∇ × X) = ∇(∇ ⋅ X) − ∇ 2 X (The last term in the right side is the vector Laplacian, not Laplacian applied on ...
The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum.It is a three-dimensional form of the wave equation.
In physics, a sigma model is a field theory that describes the field as a point particle confined to move on a fixed manifold. This manifold can be taken to be any Riemannian manifold , although it is most commonly taken to be either a Lie group or a symmetric space .
The non-linear σ-model was introduced by Gell-Mann & Lévy (1960, §6), who named it after a field corresponding to a spinless meson called σ in their model. [1] This article deals primarily with the quantization of the non-linear sigma model; please refer to the base article on the sigma model for general definitions and classical (non ...