# Tensor Analysis of the Magnetic Stress Tensor

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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