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.

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✍️Mathematical Proof

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To calculate the closed surface integral of the cross product of the position vector and the differential surface area vector, a generalized form of the divergence theorem is used. This allows the surface integral to be rewritten as a volume integral.

The generalized divergence theorem for a cross product, $\oint_S F \times d S=-\int_V(\nabla \times F) d V$, provides a method for converting a surface integral into a volume integral.

The curl of the position vector is always the zero vector ( $\nabla \times x=0$ ). This is because the partial derivatives of the independent components of the position vector with respect to each other are all zero.

As a result, the volume integral becomes the integral of the zero vector over the enclosed volume, which always evaluates to zero. This confirms that for any closed surface, the integral $\oint_S x \times d S$ is always zero.

🎬Demonstration

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).

Compare how vectors behave on a sphere and a cylinder