The analysis of the magnetic field tensor ( $F_{i j}$ ) demonstrates the power of tensor notation in physics, showing how its inherent anti-symmetry ( $F_{i j}=-F_{j i}$ ) leads directly to the symmetry of its square, $F_{i j} F_{j k}$, a necessary condition for a physical stress tensor. The derivation relies heavily on the Levi-Civita identity to compute the tensor product, yielding the key result $F_{i j} F_{j k}=B^2 \delta_{i k}-B_i B_k$, which links the fundamental magnetic field tensor to the standard vector dyadic product. Finally, by expressing the scalar field energy ( $B^2$ ) as a trace of the tensor product ( $B^2=\frac{1}{2} F_{i k} F_{k i}$ ), the entire Maxwell stress tensor ( $T_{i k}$ ) is converted into a form defined exclusively by the magnetic field tensor $F_{i j}$, ensuring mathematical consistency and demonstrating the elegance of field-based tensor formalisms.

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✍️Mathematical Proof

$\complement\cdots$Counselor

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• Anti-Symmetry and Products: The anti-symmetry of the magnetic field tensor ( $F_{i j}= -F_{j i}$ ) leads to specific symmetry properties in its products. Specifically, the product $F_{i j} F_{j k}$ is proven to be symmetric ( $F_{i j} F_{j k}=F_{k j} F_{j i}$ ), a crucial property for tensors representing physical quantities like stress.
• Contraction and the Levi-Civita Identity: The core of the calculation involves using the Levi-Civita identity to contract the product of two $\varepsilon_{i j k}$ tensors, collapsing them into combinations of Kronecker delta symbols $\left(\delta_{i k}\right)$. This is the standard method for converting products of cross-products (which $F_{i j}$ represents) into dot products and tensor products.
• Result of the Contraction: The computation $F_{i j} F_{j k}=B^2 \delta_{i k}-B_i B_k$ is an essential identity in magnetostatics. It demonstrates that the tensor product of the magnetic field tensor squared is directly related to the vector dyadic product ( $B_i B_k$ ) and the scalar magnitude squared ( $B^2$ ).
• Tensor Substitution and Consistency: The final step (part c) shows how to translate physical equations from vector notation ( $B_i$ ) to tensor notation ( $F_{i j}$ ). This process requires expressing the scalar term $B^2$ as the trace of the tensor product, $B^2=\frac{1}{2} F_{i k} F_{k i}$, ensuring the final stress tensor equation, $T_{i k}$, is internally consistent and expressed purely in terms of the fundamental field tensor $F_{i j}$.
• Maxwell Stress Tensor Structure: The final expression for the Maxwell stress tensor, $T_{i k} \propto\left(\ldots \delta_{i k}-F_{i j} F_{j k}\right)$, highlights its structure: it's composed of a term proportional to the identity tensor (representing pressure) and a term proportional to the tensor square of the field (representing tension/shear along the field lines).

✍️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

🧄Proof and Derivation-1

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