Proving Compatibility of Charts in a Union of Atlases

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In summary, the conversation is about a self-study exercise that asks to prove that the union of two atlases on a manifold is another atlas. However, the individual is unsure how to show the compatibility of charts from the two atlases. They are seeking help and suggest that there may be a typo in the exercise. There is also a suggestion that the author may have intended to ask a different question about the transitivity of compatibility between atlases.
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jojo12345
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One of the exercises in the text I'm using for self-study asks to prove that the union of a pair of atlases A and B on a manifold is another atlas. However, I don't see any way to show that two charts C,D in [tex]A\cup B[/tex] with [tex]C\in A~,~D\in B[/tex] are compatible. Could anyone give me a bit of help? Maybe just a hint?

The book is A Course in Modern Mathematical Physics by Peter Szekeres. The exercise is the first on in chapter 15.
 
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The wording of the exercise is as follows:

If A and B are two atlases on a manifold M, then their union is another atlas. Prove this statement. [Hint: A differentiable function of a differentiable function is always differentiable]

This is really verbatim from the text. However, after reading the line in the text after the exercise, I get the impression that the author made a typo. The next line:

"Any atlas can this be extended to a maximal atlas by adding to it all charts that are compatible with the charts of the atlas."

To me this suggests two things. First, the proposition in the exercise is false. Second, what the author actually wanted to ask was the following:

Show that if (1) A,B, and C are atlases on M, (2) the charts in A are compatible with the charts in B, and (3) the charts in B are compatible with the charts in C, then the charts in A are compatible with the charts in C (transitivity).

What do people think?
 

1. What is the purpose of a union of a pair of atlases?

The union of a pair of atlases is used in mathematics and physics to combine two separate coordinate systems into a single, consistent system. This is useful when studying objects or spaces that may have different coordinate systems depending on the context.

2. How is the union of a pair of atlases calculated?

The union of a pair of atlases is calculated by identifying overlapping regions between the two atlases, and then merging them to create a single, cohesive coordinate system. This process may involve transforming coordinates from one system to another using mathematical equations.

3. What is the difference between a union of a pair of atlases and a single atlas?

A single atlas is a complete and consistent coordinate system, while a union of a pair of atlases is a combination of two separate coordinate systems. The union may have overlapping regions that are defined differently in each atlas, but the overall goal is to create a single, consistent system.

4. Can the union of a pair of atlases be visualized?

Yes, the union of a pair of atlases can be visualized using graphs or diagrams that show the overlapping regions between the two coordinate systems. This can help to understand how the two systems are combined to create a single system.

5. How is the union of a pair of atlases used in scientific research?

The union of a pair of atlases is commonly used in fields such as physics and mathematics, where objects or spaces may have different coordinate systems depending on the context. It allows researchers to seamlessly combine and compare data from different sources, leading to a better understanding of complex systems.

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