Changing coordinates mean changing one set of local coordinates

In summary, a coordinate function maps local coordinates to the euclidean space for each patch on a manifold. The collection of all patches with coordinate functions covers the whole manifold. A transition function is a mapping between coordinate charts, and changing coordinates involves defining a mapping between them. There is no intrinsic coordinate system on the manifold, so we use multiple coordinate charts to cover the entire manifold.
  • #1
Ratzinger
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For each patch, a coordinate function maps the local coordinates to the euclidean space. Collection of all patches with coordinate functions covers the whole manifold.
What is a transition function and what is a change of coordinates?
Is it that you got one set of local coordinates throughout (?) the whole manifold and use a transition functions to show that the two coordinate functions are equal in the overlap regions? And does changing coordinates mean changing one set of local coordinates to another (x to x')?

Would be great if someone could help me out here. Thank you
 
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  • #2
What is a transition function and what is a change of coordinates?

A transition function is a mapping between coordinate charts. Each point of the manifold has it's own tangent space, and it's associated coordinates. You can change coordinates by defining a mapping between them.
 
  • #3
Ratzinger said:
For each patch, a coordinate function maps the local coordinates to the euclidean space.

the way that i learned it, the coordinate chart maps the points of the manifold, in that local region, to a subset of the euclidean plane R^n. there is no intrisic coordinate system on the manifold to start with.

Is it that you got one set of local coordinates throughout (?) the whole manifold and use a transition functions to show that the two

there is no way (in general) of having a coordinate system that is valid for the entire manifold - that is why we "paste" a coordinate system onto a small region around a particular point. however, as long as we are able to do the same thing for a different overlapping region, then we can repeat this process over the whole manifold. (the collection of all of these coordinate charts for the whole manifold is called the atlas.)

consider the 2-sphere for example, and make your coordinate charts by projecting various hemispheres onto a plane, from all six axial positions, and you'll see what I mean - projecting the "western" hemisphere onto an adjacent plane will be 1-to-1, but then couldn't include the "eastern" (opposite side) hemisphere, that would require yet another plane (aka coordinate chart).

hope this clears things up a bit.
 

1. What are local coordinates?

Local coordinates refer to a specific system of measuring and describing points in space. They are typically used to describe the position or orientation of objects or systems within a larger coordinate system.

2. Why would you need to change local coordinates?

There are several reasons why one might need to change local coordinates. For example, if you are working with a different coordinate system, or if you need to translate or rotate an object in space, you may need to change the local coordinates to accurately describe its position or orientation.

3. How do you change local coordinates?

The process of changing local coordinates will vary depending on the specific situation. In general, it involves using mathematical transformations such as translation, rotation, or scaling to convert the coordinates from one system to another. This may involve using equations or software programs to perform the necessary calculations.

4. What are the potential challenges of changing local coordinates?

One challenge of changing local coordinates is ensuring accuracy and precision in the calculations. Any errors or inconsistencies can result in incorrect positioning or orientation of objects. Additionally, if the coordinate systems being used are vastly different, it may be difficult to accurately translate between them.

5. How does changing local coordinates impact scientific research?

Changing local coordinates is an essential part of many scientific studies, particularly in fields such as physics, engineering, and computer science. It allows researchers to accurately analyze and visualize data, make predictions, and design experiments. Without the ability to change local coordinates, many scientific studies and advancements would not be possible.

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