Electric susceptibility is a rank two tensor, proving its tensorial nature by showing its components transform according to the mixed-tensor transformation law, which explains how an electric field can induce polarization in a different direction in an anisotropic medium.

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✍️Mathematical Proof

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Anisotropic Behavior: Unlike in isotropic media where the relationship between polarization ( $P$ ) and the electric field ( $E$ ) is a simple scalar product, in an anisotropic medium, the two vectors are still linearly related, but the relationship is described by a more complex quantity. This is because the medium's response to an electric field depends on the field's direction.

Electric Susceptibility as a Tensor: The electric susceptibility, $\chi$, is not a simple scalar constant in an anisotropic medium. Instead, it must be a rank two tensor $\left(\chi_j^i\right)$. This tensor's components account for how an electric field in one direction can induce a polarization in a potentially different direction.

The Role of the Tensor Transformation Law: The proof that $\chi_j^i$ is a tensor relies on the tensor transformation law. By showing that the relationship $P^i=\varepsilon_0 \chi_j^i E^j$ holds true in any coordinate system - with $\chi_j^i$ transforming as a mixed-rank two tensor-we confirm its tensorial nature. This invariance under coordinate transformations is the defining characteristic of a tensor.

Implications for Physics: Understanding susceptibility as a tensor is crucial in fields like optics and condensed matter physics. It explains phenomena such as birefringence, where a single light ray splits into two due to the anisotropic optical properties of a crystal, and demonstrates the power of tensor analysis in describing complex physical phenomena.

✍️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

🧄Proof and Derivation-1

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