This study investigates the evaluation of the surface integral of a position vector cross product, defined as $\vec{I}=\oint_S \vec{x} \times d \vec{S}$. Utilizing a generalized divergence theorem, the analysis demonstrates that this surface integral can be transformed into a volume integral of the curl of the position vector. Mathematical derivation proves that the curl of the position vector is the zero vector, leading to the conclusion that the integral over any closed surface is always zero.
The research extends into the behavior of open surfaces, where the result is generally non-zero and depends on the geometry of the boundary curve. By converting the surface integral into a line integral around the boundary, the study illustrates how rotational symmetry influences the outcome. Comparative demonstrations of a centered disk and a shifted hemisphere reveal that breaking symmetry relative to the origin creates a "leverage" imbalance. Physically, the integral is interpreted as the net torque exerted by uniform pressure; while closed or centered surfaces maintain static equilibrium, shifted open surfaces experience a net twisting force.
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