# What is a surface in Maths and/or Physics?

• 11-16-2017, 10:49 AM
geordief
What is a surface in Maths and/or Physics?
I have learned in another forum that a ID lines can be a surface on a 2D plane and a 2D plane can be a surface on a 3D Volume.

Can this be generalized mathematically as we introduce more (higher) dimensions?

Is it the case (as it seems to me to be true in the examples I have given) that when we set one of the dimensional variables to a constant that what we have is a surface ?(I think they may be referred to as hypersurfaces in higher dimensions)

By setting one of the dimensional variables to a constant (decreasing its value range to smaller and smaller quantities in the same way as is done in Calculus *,do we effectively transform the n-Dimensional model to a n-1 Dimensional model and create a (hyper)surface that "straddles" the 2 models?

*ie as a limit
• 11-17-2017, 05:01 AM
AndrewC
Quote:

Originally Posted by geordief
I have learned in another forum that a ID lines can be a surface on a 2D plane and a 2D plane can be a surface on a 3D Volume.

Can this be generalized mathematically as we introduce more (higher) dimensions?

Is it the case (as it seems to me to be true in the examples I have given) that when we set one of the dimensional variables to a constant that what we have is a surface ?(I think they may be referred to as hypersurfaces in higher dimensions)

By setting one of the dimensional variables to a constant (decreasing its value range to smaller and smaller quantities in the same way as is done in Calculus *,do we effectively transform the n-Dimensional model to a n-1 Dimensional model and create a (hyper)surface that "straddles" the 2 models?

*ie as a limit

Here is the rigorous mathematical treatment of the concept called "embedding".
• 01-07-2018, 10:12 AM
geordief
I am still not clear if a hypersurface is 2 dimensional or [n-1] dimensional for a n-dimensional space.

In particular in 4-D space-time does one work with 2-D surfaces (for example in calculating curvature by means of parallel transporting of a vector) or is use also made of 3-D surfaces (presumably ,at least in my mind constructed by setting one of the dimensional variables to a constant).

If the 3-D surface does exist and is useful can it be built up successively from the 6(?) combinations of 2-D at any point (event)?
• 01-07-2018, 04:26 PM
AndrewC
Quote:

Originally Posted by geordief
I am still not clear if a hypersurface is 2 dimensional

Only in 3 dimensions

Quote:

or [n-1] dimensional for a n-dimensional space.
The math is quite clear on the subject. Have you read the web page I linked for you?
• 01-07-2018, 11:46 PM
geordief
No but I will now. :o
And I'll get back then......
• 01-08-2018, 02:11 AM
geordief
Quote:

Originally Posted by AndrewC
Only in 3 dimensions

The math is quite clear on the subject. Have you read the web page I linked for you?

OK I have read the main body of that link and think I understand that a hypersurface is embedded in the manifold with 1 more dimensions than it has.

Am I right though to think that there are nevertheless also 2D surfaces in higher dimensional manifolds (esp 4D space-time manifolds) even if is incorrect to refer to them as "hypersurfaces".

Are they just referred to as surfaces or 2D surfaces?
• 01-08-2018, 03:27 PM
AndrewC
Quote:

Originally Posted by geordief
OK I have read the main body of that link and think I understand that a hypersurface is embedded in the manifold with 1 more dimensions than it has.

Am I right though to think that there are nevertheless also 2D surfaces in higher dimensional manifolds (esp 4D space-time manifolds) even if is incorrect to refer to them as "hypersurfaces".

Are they just referred to as surfaces or 2D surfaces?

yes