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JYM
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I want to show that a closed unit ball is manifold with boundary and I attempted as uploaded. But I am not happy with the way I showed the boundary chart is injective. Am I right?
A manifold with a boundary is a mathematical object that is locally similar to Euclidean space, but may have a boundary. It is a type of topological space that is smooth and can be represented by a finite number of coordinate systems.
The main difference between a manifold and a manifold with a boundary is the presence of a boundary. A manifold has no boundary, while a manifold with a boundary has at least one boundary component. Additionally, the behavior of a manifold with a boundary near the boundary is different from its behavior in the interior.
A manifold with a boundary is typically represented using a coordinate system, where each point on the manifold is assigned a set of coordinates. These coordinates are used to define the shape and properties of the manifold, including its boundary.
Some common examples of manifolds with a boundary include a sphere with a boundary (also known as a ball), a cylinder, and a cone. In each of these examples, the boundary is a lower-dimensional object (a point, circle, and line, respectively) that separates the interior from the exterior of the manifold.
Manifolds with a boundary are used in various areas of mathematics and physics, particularly in differential geometry and topology. They are also used in computer graphics and computer vision to represent and analyze shapes and structures in a more efficient and accurate way.