The verification confirms that the single four-dimensional tensor equation $\partial_\mu F^{\mu \nu}=K^\nu$ successfully unifies two of Maxwell's equations. By analyzing the $\nu=0$ (time) component, we derive Gauss's Law ( $\nabla \cdot E =\rho / \varepsilon_0$ ), which relates the divergence of the electric field to the charge density. By analyzing the $\nu=j$ (spatial) components, we derive the Ampère-Maxwell Law ( $\nabla \times B=\mu_0 J+\mu_0 \varepsilon_0 \frac{\partial E}{\partial t}$ ), which relates the curl of the magnetic field to both the current density and the time rate of change of the electric field. These two are collectively known as the inhomogeneous Maxwell's equations because they are sourced by the charge and current densities embedded in the four-vector $K^\nu$, demonstrating the elegant and compact nature of electromagnetism within the framework of special relativity.

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✍️Mathematical Proof

$\complement\cdots$Counselor

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• Tensor Notation Unifies Equations: The single four-dimensional tensor equation $\partial_\mu F^{\mu \nu}=K^\nu$ elegantly combines two of Maxwell's four equations: Gauss's Law and the Ampère-Maxwell Law.
• Inhomogeneous Equations: The equations derived-Gauss's Law ( $\nabla \cdot E =\frac{\rho}{\varepsilon_0}$ ) and the Ampère-Maxwell Law ( $\nabla \times B=\mu_0 J+\mu_0 \varepsilon_0 \frac{\partial E}{\partial t}$ )-are the inhomogeneous Maxwell's equations, which relate the fields ( $E$ and $B$ ) directly to their sources (charge density $\rho$ and current density $J$ ).
• Time Component yields Gauss's Law: Setting the free index $\nu=0$ (the time component) results in $\nabla \cdot E=\frac{\rho}{\varepsilon_0}$.
• Spatial Components yield Ampère-Maxwell Law: Setting the free index $\nu=j$ (the spatial components) results in the vector equation $\nabla \times B=\mu_0 J+\mu_0 \varepsilon_0 \frac{\partial E}{\partial t}$.
• Role of $F^{\mu \nu}$ and $K^\mu$ :
  • The electromagnetic field tensor ( $F^{\mu \nu}$ ) successfully embeds both the $E$ and $B$ fields into a single anti-symmetric object.
  • The four-current density ( $K^\mu$ ) embeds both the charge density ( $\rho$ ) and the current density ( $J$ ) into a single four-vector.
• Relativistic Foundation: This formulation uses the four-dimensional spacetime coordinate $x^\mu=\left(c t, x^1, x^2, x^3\right)$ and is the natural way to write Maxwell's equations, demonstrating that they are inherently consistent with the principles of Special Relativity.

✍️Mathematical Proof

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1. Derivation of Tensor Transformation Properties for Mixed Tensors
2. The Polar Tensor Basis in Cartesian Form
3. Verifying the Rank Two Zero Tensor
4. Tensor Analysis of Electric Susceptibility in Anisotropic Media
5. Analysis of Ohm's Law in an Anisotropic Medium
6. Verifying Tensor Transformations
7. Proof of Coordinate Independence of Tensor Contraction
8. Proof of a Tensor's Invariance Property
9. Proving Symmetry of a Rank-2 Tensor
10. Tensor Symmetrization and Anti-Symmetrization Properties
11. Symmetric and Antisymmetric Tensor Contractions
12. The Uniqueness of the Zero Tensor under Specific Symmetry Constraints
13. Counting Independent Tensor Components Based on Symmetry
14. Transformation of the Inverse Metric Tensor
15. Finding the Covariant Components of a Magnetic Field
16. Covariant Nature of the Gradient
17. Christoffel Symbol Transformation Rule Derivation
18. Contraction of the Christoffel Symbols and the Metric Determinant
19. Divergence of an Antisymmetric Tensor in Terms of the Metric Determinant
20. Calculation of the Metric Tensor and Christoffel Symbols in Spherical Coordinates
21. Christoffel Symbols for Cylindrical Coordinates
22. Finding Arc Length and Curve Length in Spherical Coordinates
23. Solving for Metric Tensors and Christoffel Symbols
24. Metric Tensor and Line Element in Non-Orthogonal Coordinates
25. Tensor vs. Non-Tensor Transformation of Derivatives
26. Verification of Covariant Derivative Identities
27. Divergence in Spherical Coordinates Derivation and Verification
28. Laplace Operator Derivation and Verification in Cylindrical Coordinates
29. Divergence of Tangent Basis Vectors in Curvilinear Coordinates
30. Derivation of the Laplacian Operator in General Curvilinear Coordinates
31. Verification of Tensor Density Operations
32. Verification of the Product Rule for Jacobian Determinants and Tensor Density Transformation
33. Metric Determinant and Cross Product in Scaled Coordinates
34. Vanishing Divergence of the Levi-Civita Tensor
35. Curl and Vector Cross-Product Identity in General Coordinates
36. Curl of the Dual Basis in Cylindrical and Spherical Coordinates
37. Proof of Covariant Index Anti-Symmetrisation
38. Affine Transformations and the Orthogonality of Cartesian Rotations
39. Fluid Mechanics Integrals for Mass and Motion
40. Volume Elements in Non-Cartesian Coordinates (Jacobian Method)
41. Young's Modulus and Poisson's Ratio in Terms of Bulk and Shear Moduli
42. Tensor Analysis of the Magnetic Stress Tensor
43. Surface Force for Two Equal Charges
44. Total Electromagnetic Force in a Source-Free Static Volume
45. Proof of the Rotational Identity
46. Finding the Generalized Inertia Tensor for the Coupled Mass System
47. Tensor Form of the Centrifugal Force in Rotating Frames
48. Derivation and Calculation of the Gravitational Tidal Tensor
49. Conversion of Total Magnetic Force to a Surface Integral via the Maxwell Stress Tensor
50. Verifying the Inhomogeneous Maxwell's Equations in Spacetime

🧄Proof and Derivation-1

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