Smoothness of surfaces

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In summary, according to Presley's book "Elementary Differential Geometry", a smooth surface is a surface that has an atlas consisting of regular surface patches. The atlas of a surface is a collection of homeomorphisms that cover it, and a surface patch is a homeomorphism in the atlas. A patch is regular if it is smooth and its first partial derivatives are linearly independent at all points on its domain. Generally, there can be multiple atlases for a given surface and smoothness is not an intrinsic property, but rather dependent on the choice of parametrization. In the book, it is proven that if a subset of R^3 has certain properties, it is a smooth surface under a suitable atlas. The author defines
  • #1
quasar987
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According to Presley (Elementary differential geometry),

"A smooth surface is a surface [itex]\mathbf{\sigma}[/itex] whose atlas consists of regular surface patches."

(The atlas of a surface is a collection of homeomorphisms that "cover" it. A surface patch is just another word for an homeomorphism in the atlas. Finally, a surface patch is regular if it is smooth and its first partial derivatives are linearly independant at all points (u,v) of its domain.)

Generally, there are several possible distinct atlases for a given surface. A priori, I see no reason to say that if a surface is smooth under some atlas, it is under every atlas.

So, is it really so that smoothness is not an intrinsic property of surfaces, but rather a "bonus" that comes with a proper choice of parametrization? (much like regularness of a curve is a property of the parametrization, not of the trace itself)
 
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  • #2
Yes, if I recall correctly, it depends on the atlas.
 
  • #3
But then there is a thm latter on that goes

Let S be a subset of R^3 with the following property: for each point P in S, there is an open subset W of R^3 containging P and a smooth function f:W-->R such that
(i) S [itex]\cap[/itex] W={(x,y,z) in W: f(x,y,z)=c};
(ii) the partial derivatives f_x, f_y and f_z do not all vanish at P.
Then S is a smooth surface.


The statement "S is a smooth surface" doesn't make sense on its own! One that does however is "Then there is an atlas for which S is smooth". The way the author proves this thm is by finding an atlas of regular patches, i.e. one for which S is smooth. So it would appear that this is all the author meant by "S is smooth". Unless, we've overlooked something and smoothness really is an intrinsic property!
 
  • #4
first you have to define smoothness. when you do, you will see that it depends eitheron the parametrization by surfqce patches, or on the embedding, which also tacitly ssumes that projection on some axes is an allowable family of paTCHES.
 
  • #5
BY THE WAY, the imprecision of those descriptions made me assume you were reading a physics book and not a math book. gosh. i recommend geting a better book. like do carmo.or spivak, or shifrins web notes, or if those are too advanced, maybe barett o'neill for a baby book.
 
  • #6
It's a maths book. Elementary Geometry by Andrew Pressley.

But thanks for the books recommendation!

Which spivak book are you reffering to?
 

1. What is considered a smooth surface?

A smooth surface is one that has very little variation or irregularities in its texture or appearance. It is typically described as being flat, even, and free of bumps or roughness.

2. How is the smoothness of a surface measured?

The smoothness of a surface can be measured using different methods, depending on the type of surface. One common way is to use a surface roughness tester, which uses a probe to scan the surface and measure its roughness parameters such as Ra (average roughness) and Rz (maximum roughness).

3. Why is smoothness important in certain industries?

Smoothness is important in industries such as manufacturing and engineering because it can affect the performance, functionality, and aesthetics of products. In industries like aerospace and automotive, smooth surfaces are essential for reducing drag and improving fuel efficiency.

4. What are some common factors that can affect the smoothness of a surface?

The smoothness of a surface can be affected by factors such as the material used, the method of production or manufacturing, and environmental conditions like temperature and humidity. Other factors like wear and tear, corrosion, and surface treatments can also impact the smoothness of a surface over time.

5. How can the smoothness of a surface be improved?

The smoothness of a surface can be improved through various methods, including sanding, polishing, grinding, and chemical treatments. Using high-quality materials and precise manufacturing techniques can also help achieve a smoother surface. In some cases, adding a coating or finish can also improve the smoothness of a surface.

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