Euclidean space is the natural environment for Calculus, but manifolds allow us to extend Calculus to curved spaces.
We’ve been working with manifolds since the first article in this series, but we didn’t say we were working with manifolds until An Intro to Differential Geometry. Even then, we only said that a manifold had to be locally Euclidean, but that’s only one of three requirements we want of manifolds.
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More specifically, we’re going to want a manifold to be a mathematical space
- that’s locally Euclidean,
- that has unique limits,
- and that we can integrate over.
If we have a space that satisfies all these conditions, we’ll have a manifold.
Check Your Understanding
There’s not too much to do here because I’m trying to explain why manifolds are defined as they are. If you want more questions, look up all the bold terms in this article and read what you find. You can also go through some of the resources linked in the Further Reading section.
Spaces That Aren’t Manifolds
None of the following spaces are manifolds.
- A figure eight
- A cone
- The Cantor set
- The union of a line and a disk (filled circle) where the line intersects the disk.
- The particular point topology, where the open sets consist of all sets that contain the point p.
- The Rational Numbers
- A sphere with hair.
Partitions of Unity
Prove that the integration on a manifold does not depend on the specific partition of unity you choose.