Abstract
In this chapter we introduce and describe the concepts of surface, boundary of a surface, coherent orientations of a surface and its boundary. We derive a formula for calculating the area of a surface, and we give an initial idea of the concept of a differential form. All these concepts are essential when working with curvilinear and surface integrals, which will be the main subject of the next chapter.
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Notes
- 1.
As before, a neighborhood of a point \(x\in S\subset\mathbb{R}^{n}\) in \(S\) is a set Us \((x)=S\cap U(x)\), where \(U(x)\) is a neighborhood of \(x\) in \(\mathbb{R}^{n}\). Since we shall be discussing only neighborhoods of a point on a surface in what follows, we shall simplify the notation where no confusion can arise by writing \(U\) or \(U(x)\) instead of \(U_{S}(x)\).
- 2.
On \(S\subset\mathbb{R}^{n}\) and hence also on \(U\subset S\) there is a unique metric induced from \(\mathbb{R}^{n}\), so that one can speak of a topological mapping of \(U\) into \(\mathbb{R}^{k}\).
- 3.
An example of the surface described here was constructed by the American topologist J.W. Alexander (1888–1977).
- 4.
For the tangent plane see Sect. 8.7.
- 5.
A.F. Möbius (1790–1868) – German mathematician and astronomer.
- 6.
F.Ch. Klein (1849–1925) – outstanding German mathematician, the first to make a rigorous investigation of non-Euclidean geometry. An expert in the history of mathematics and one of the organizers of the “Encyclopädie der mathematischen Wisaenschaftm”.
- 7.
H. Whitney (1907–1989) – American topologist, one of the founders of the theory of fiber bundles.
- 8.
See the footnote on p. 497.
- 9.
If (12.29) is used pointwise, one can see that
$$ \varphi^{\ast} \bigl(a(x)\omega\bigr)=a \bigl(\varphi(t) \bigr) \varphi^{\ast}\omega. $$
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Zorich, V.A. (2016). Surfaces and Differential Forms in \(\mathbb{R}^{n}\) . In: Mathematical Analysis II. Universitext. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48993-2_4
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DOI: https://doi.org/10.1007/978-3-662-48993-2_4
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