= Multiplicity of a parametrized surface integral
{title2=$\int_U f(\mathbf r(u,v))|\mathbf r_u\times\mathbf r_v|\,du\,dv$}
For a regular <parametric surface> $\mathbf r:U\to X$, the displayed <surface integral> counts each image point as many times as its preimages in $U$, apart from sets of zero <surface area>. It equals the ordinary integral over the image when the parametrization is <injective> except on such sets. A periodic parametrization integrated over more than one period can give a different answer from integration over its image set. For example, $(u,v)\mapsto(\cos u,\sin u,v)$ with $0\leq u\leq4\pi$ and $0\leq v\leq1$ gives parameter area $4\pi$, whereas the image <circular cylinder> has <surface area> $2\pi$.
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