Editing Surface area (section)

==Definition==
While the areas of many simple surfaces have been known since antiquity, a rigorous mathematical ''definition'' of area requires a great deal of care.
This should provide a function

: <math> S \mapsto A(S) </math>

which assigns a positive [[real number]] to a certain class of [[Surface (topology)|surface]]s that satisfies several natural requirements. The most fundamental property of the surface area is its '''additivity''': ''the area of the whole is the sum of the areas of the parts''. More rigorously, if a surface ''S'' is a union of finitely many pieces ''S''<sub>1</sub>, …, ''S''<sub>''r''</sub> which do not overlap except at their boundaries, then 
: <math> A(S) = A(S_1) + \cdots + A(S_r). </math>

Surface areas of flat polygonal shapes must agree with their geometrically defined [[area]]. Since surface area is a geometric notion, areas of [[congruence (geometry)|congruent]] surfaces must be the same and the area must depend only on the shape of the surface, but not on its position and orientation in space. This means that surface area is invariant under the [[Euclidean group|group of Euclidean motions]]. These properties uniquely characterize surface area for a wide class of geometric surfaces called ''piecewise smooth''. Such surfaces consist of finitely many pieces that can be represented in the [[parametric surface|parametric form]]

: <math> S_D: \vec{r}=\vec{r}(u,v), \quad (u,v)\in D </math>

with a [[continuously differentiable]] function <math>\vec{r}.</math> The area of an individual piece is defined by the formula

: <math> A(S_D) = \iint_D\left |\vec{r}_u\times\vec{r}_v\right | \, du \, dv. </math>

Thus the area of ''S''<sub>''D''</sub> is obtained by integrating the length of the normal vector <math>\vec{r}_u\times\vec{r}_v</math> to the surface over the appropriate region ''D'' in the parametric ''uv'' plane. The area of the whole surface is then obtained by adding together the areas of the pieces, using additivity of surface area. The main formula can be specialized to different classes of surfaces, giving, in particular, formulas for areas of graphs ''z'' = ''f''(''x'',''y'') and [[surface of revolution|surfaces of revolution]].

[[File:Schwarz-lantern.gif|thumb|[[Schwarz lantern]] with <math>M</math> axial slices and <math>N</math> radial vertices. The limit of the area as <math>M</math> and <math>N</math> tend to infinity doesn't converge. In particular it doesn't converge to the area of the cylinder.]]One of the subtleties of surface area, as compared to [[arc length]] of curves, is that surface area cannot be defined simply as the limit of areas of polyhedral shapes approximating a given smooth surface. It was demonstrated by [[Hermann Schwarz]] that already for the cylinder, different choices of approximating flat surfaces can lead to different limiting values of the area; this example is known as the [[Schwarz lantern]].<ref name=sch1>{{cite web|url=http://fredrickey.info/hm/CalcNotes/schwarz-paradox.pdf|title=Schwarz's Paradox|access-date=2017-03-21|url-status=live|archive-url=https://web.archive.org/web/20160304073957/http://fredrickey.info/hm/CalcNotes/schwarz-paradox.pdf|archive-date=2016-03-04}}</ref><ref name=sch2>{{cite web |url=http://mathdl.maa.org/images/upload_library/22/Polya/00494925.di020678.02p0385w.pdf |title=Archived copy |access-date=2012-07-24 |url-status=dead |archive-url=https://web.archive.org/web/20111215152255/http://mathdl.maa.org/images/upload_library/22/Polya/00494925.di020678.02p0385w.pdf |archive-date=2011-12-15 }}</ref>

Various approaches to a general definition of surface area were developed in the late nineteenth and the early twentieth century by [[Henri Lebesgue]] and [[Hermann Minkowski]]. While for piecewise smooth surfaces there is a unique natural notion of surface area, if a surface is very irregular, or rough, then it may not be possible to assign an area to it at all. A typical example is given by a surface with spikes spread throughout in a dense fashion. Many surfaces of this type occur in the study of [[fractal]]s. Extensions of the notion of area which partially fulfill its function and may be defined even for very badly irregular surfaces are studied in [[geometric measure theory]]. A specific example of such an extension is the [[Minkowski content]] of the surface.