# Conversion of Total Magnetic Force to a Surface Integral via the Maxwell Stress Tensor

Transforming the magnetic force integral is the conceptual shift from action-at-a-distance to the idea that the force is conveyed entirely by the Magnetic Stress Tensor ( $$T ^{ i j }$$ ) acting on the boundary surface. This rank-two tensor describes the momentum flux exerted by the magnetic field across that boundary, allowing the total force to be calculated simply by measuring the field B on the surface S. Physically, the tensor's components reveal the dual nature of these field mediated stresses: the off-diagonal terms ( $$B\_i B\_j$$ ) represent shear stresses that pull the surface diagonally, while the diagonal terms ( $$\frac{1}{2} \delta\_{i j} B^2$$ ) represent a uniform outward pressure exerted perpendicular to the field lines.

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