The derivation outlines the process of converting the total magnetic force ($\vec{F}$) acting on a volume $V$, initially expressed as a volume integral dependent on the internal current density ($\vec{F}=\int_V(\jmath \times B) d V$), into a surface integral involving only the magnetic field ($\vec{B}$) on the boundary surface $S$. This conversion requires that the magnetic field satisfies the Maxwell conditions $\nabla \cdot \vec{B}=0$ and $\nabla \times \vec{B}=\mu_0 \vec{\jmath}$. By substituting $\vec{\jmath}$ using Maxwell's equation, applying vector identities facilitated by $\nabla \cdot B=0$, and manipulating the components using the Kronecker delta ($\delta_{i j}$), the force component $F_i$ is recast as the divergence of a tensor field. Finally, the Divergence Theorem (Gauss's Theorem) is applied to transform this volume integral into the required surface integral, yielding the result $\vec{F}=\oint_S \vec{e}i T^{i j} d S_j$, where $T^{i j}$ is the rank two Magnetic Stress Tensor with components $T^{i j}=\frac{1}{\mu_0}\left(B_i B_j-\frac{1}{2} \delta{i j} B^2\right)$, describing the momentum flux per unit area across the surface $S$.

From Currents to Surfaces Unlocking the Total Magnetic Force with the Maxwell Stress Tensor-L.mp4