Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics IV / 3 / a

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Solution

For a regular parametrized surface that covers its image once, up to sets of zero area, its surface area is

Area (S) = \iint_{U} \frac{\partial r}{\partial u} \times \frac{\partial r}{\partial v} d u d v .

The integrand is the area element of a surface, the area of the small parallelogram spanned by the two tangent vectors. A parametrization that covers a region more than once counts that multiplicity; the geometric surface area requires a single covering or a division into nonoverlapping patches.

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