# 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. ### :clapper: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). {% embed url="" %} ### :pen\_ballpoint:Mathematical Proof {% embed url="" %}