🧄Surface Integral to Volume Integral Conversion Using the Divergence Theorem

The closed surface integral $\oint_S x \times d S$ is always zero because the curl of the position vector ( $\nabla \times x$ ) is always zero, a mathematical result that is physically consistent with vectors either being individually zero or cancelling each other out due to a surface's symmetry.

🎬Compare how vectors behave on a sphere and a cylinder

A zero result for a surface integral can be achieved either because all individual vectors are zero (as seen on the sphere), or because non-zero vectors cancel each other out due to symmetry (as seen on the cylinder).

🖊️Mathematical Proof

Surface Integral to Volume Integral Conversion Using the Divergence Theorem | Proof and Derivationviadean on Notion