How is the region between two concentric spheres simply connected?

  • Thread starter quasar_4
  • Start date
In summary, the conversation discusses the concept of a simply connected region, specifically the region between two concentric spheres. The book being read mentions that this region is simply connected, but the individual is having trouble understanding how. It is explained that the curve can shrink to a point by sliding along the surface of the interior sphere, rather than just sitting on the equator. This concept also applies to a spiral surface, where the curve can be drawn anywhere and then shrink by sliding down the spiral.
  • #1
quasar_4
290
0
Hi,

I've been reviewing multivariable calculus, which I took ages ago, and trying to understand the concept of a simply connected region. The book I'm reading discusses how the region between two concentric spheres is simply connected, but I'm having trouble seeing it. If I think about that region, and imagine sticking a curve there and shrinking it down, don't I run into trouble when I hit the boundary of interior sphere? Can anyone explain to me how this region is simply connected?

Thanks.
 
Physics news on Phys.org
  • #2
Once the curve hits the boundary of the interior sphere it can simply slide along that surface to shrink to a point. It doesn't have to just sit at an equator. That extra dimension in which to shrink is what makes it different from the space between two concentric circles in the plane.
 
  • #3
Ok, I see. I was trying to restrict it to a given equator. The same idea then applies to the spiral surface - you can draw your closed curve anywhere on the spiral and let it slide down the spiral to shrink it, right?
 
  • #4
Yes.
 

Related to How is the region between two concentric spheres simply connected?

1. What does it mean for a region to be "simply connected"?

A simply connected region is a type of topological space that is connected and does not have any "holes" or "handles". In other words, any loop or closed curve in the region can be continuously shrunk to a point without leaving the region. This means that the region is topologically equivalent to a disk.

2. How can one determine if a region is simply connected?

One way to determine if a region is simply connected is to check if any loop or closed curve in the region can be continuously shrunk to a point without leaving the region. Another way is to use the Riemann mapping theorem, which states that any simply connected region is conformally equivalent to the unit disk.

3. Are simply connected regions important in mathematics?

Yes, simply connected regions have many applications in mathematics, particularly in complex analysis and topology. For example, they are used in the proof of the fundamental theorem of algebra, which states that any non-constant polynomial with complex coefficients has at least one complex root. They are also important in the study of surfaces and their properties.

4. Can a region be simply connected but not connected?

No, a region must be both connected and have no holes in order to be considered simply connected. If a region is not connected, it means that it can be divided into two or more disconnected parts, which would not allow for any closed curves to be shrunk to a point without leaving the region.

5. How are simply connected regions related to contractible spaces?

Simply connected regions are a type of contractible space, which means that they can be continuously deformed to a single point without changing their topological properties. However, not all contractible spaces are simply connected, as there are other types of contractible spaces that may have holes or other more complicated topological structures.

Similar threads

I Is Being Path-Connected the Key to Understanding Simply Connected Regions?
    • Calculus
Replies
6
Views
2K
I Diverging Gaussian curvature and (non) simply connected regions
    • Differential Geometry
Replies
1
Views
2K
I must understand connectivity wrong, because my book says this. The
    • Calculus
Replies
2
Views
2K
Help, can't visualize a fantasy worldbuilding mechanic
    • Sci-Fi Writing and World Building
Replies
1
Views
652
I Rigorous: Where is the Quantum System Prior to Measurement?
    • Quantum Interpretations and Foundations
Replies
12
Views
952
How Do Charges Distribute Between Concentric Spheres When Connected by a Wire?
    • Introductory Physics Homework Help
Replies
3
Views
1K
Finding volume of a modified sphere using integrals
    • Calculus
Replies
3
Views
2K
Gauss Law: Concentric Spheres
    • Introductory Physics Homework Help
Replies
8
Views
3K
Capacitance and field of 2 eccentric spheres
    • Advanced Physics Homework Help
Replies
3
Views
2K
Is Any Closed Curve Homotopic to a Constant Curve in a Simply Connected Region?
    • Calculus
Replies
3
Views
3K

Hot Threads

Recent Insights

Back
Top