🎬how the magnetic stress tensor decomposes to show that magnetic fields simultaneously exert tension

The visualizations powerfully confirm that the magnetic field tensor product ( $F_{i j} F_{j k}$ ) and the Maxwell Stress Tensor ( $T_{i k}$ ) are defined by the interplay between isotropic energy and field direction. The tensor product $F_{i j} F_{j k}$ is constructed by subtracting the highly directional dyadic product $\left(B_i B_k\right)$ from the uniform isotropic term $\left(B^2 \delta_{i k}\right)$. This leads to the fundamental physical interpretation of $T_{i k}$, which is composed of the dyadic term (representing tension/pulling force along the field lines) and the isotropic term (representing pressure perpendicular to the field lines). When the components are summed, the resulting $T_{i k}$ tensor is always positive along the field direction (net tension) and negative perpendicular to the field direction (net pressure/compression), precisely illustrating the classic physical effect of magnetic fields on surrounding media.

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