🧄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.

Tensor Analysis of the Magnetic Stress Tensor | Proof and Derivationviadean on Notion