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Definition of surface area

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This is exactly the definition of surface area that was given in the leading Soviet school geometry textbook of the 1980s, by the author Pogorelov, in grade 11 (except that volume was not defined as Lebesgue measure).
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You can. But I forget what it’s called.
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For nice surfaces there are several equivalent ways to calculate area: parametrization, Minkowski content, Hausdorff measure. It seems for bad surfaces Hausdorff measure is usually better than Minkowski content
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One problem is that Gabriel's Horn would have infinite surface area by that definition (which would violate sigma-additivity). You can get around this, but you are substituting some technical difficulties for others, not completely getting rid of them.
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Hi, I am probably going to sound like one of those old people who type Google searches into facebook .... I now understand your question, you are asking about how to induce a metric from an existing embedding of a smooth real surface into Euclidean space. If you start with a surface which is not embedded into Euclidean space the Riemannian metrics form an uncountable set that is easy to calculate; you are dealing with how to choose one particular Riemann metric. And there is no issue with how any Riemannian metric has an underlying two-form whose integral gives the area. Also if you interpret a Riemannian metric as an element of the second symmetric power of differential one-forms, the conormal bundle has a natural embedding into the one-forms of the Euclidean space restricted to the surface, and the quotient is isomorphic to the one-forms of the surface. You apply the induced map on second symmetric powers to the element which defines the Riemann metric of the Euclidean space, and this gives an element of the second symmetric power of the one-forms of the surface, which is your Riemannian metric of the surface.

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