# 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.

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