dextercioby said:That's a set of Maxwell's equations written in 4-tensor form. Which Bianchi's identities do you mean? The ones with the Riemann curvature
Maxwell's Homogeneous Equations are a set of four partial differential equations that describe the behavior of electric and magnetic fields in a vacuum. They were developed by James Clerk Maxwell in the 19th century and are fundamental to the study of electromagnetism.
The notation used in Maxwell's Homogeneous Equations is the vector calculus notation, which includes the use of the nabla symbol (∇) and the cross product (×). The equations also use the symbols for electric field (E), magnetic field (B), charge density (ρ), and current density (J).
The first equation, also known as Gauss's Law, relates the divergence of the electric field to the charge density. The second equation, known as Gauss's Law for magnetism, states that there are no magnetic monopoles and relates the divergence of the magnetic field to zero. The third equation, known as Faraday's Law, relates the curl of the electric field to the time rate of change of the magnetic field. The fourth equation, known as Ampere's Law, relates the curl of the magnetic field to the sum of the time rate of change of the electric field and the current density.
Maxwell's Homogeneous Equations are used in many practical applications, including the design of electronic devices, the development of telecommunications technologies, and the study of electromagnetic waves. They are also used in the development of theories and laws in areas such as optics and quantum mechanics.
Maxwell's Homogeneous Equations, along with Maxwell's Inhomogeneous Equations, make up the complete set of Maxwell's Equations, which describe the behavior of electric and magnetic fields in the presence of sources (charges and currents). They are also related to other fundamental equations of electromagnetism, such as Coulomb's Law, the Biot-Savart Law, and Ohm's Law.