Differentiable function - definition on a manifold

In summary, the conversation discusses the definition of differentiable functions on a differential manifold using charts and atlases. The question posed is whether a function that is not differentiable in one chart can become differentiable when represented in another chart. The answer is that if a function is not differentiable in one chart, it will not be differentiable in any other chart according to the compatibility of differentiable functions across charts in the same differential structure.
  • #1
cianfa72
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Hi,

a basic question related to differential manifold definition.

Leveraging on the atlas's charts ##\left\{(U_i,\varphi_i)\right\} ## we actually define on ##M## the notion of differentiable function. Now take a specific chart ##\left(U,\varphi \right)## and consider a function ##f## defined on it that happens to be not differentiable in that specific chart.

My question is: for the given function defined on ##M## could it be the case in resulting differentiable when represented in another atlas's chart (e.g. in the ##\left(V,\gamma \right)## chart supposed to be compatible with ##\left(U,\varphi \right)## )?

In other words: we know the notion of differentiable function is compatible across charts belonging to the same differential structure, but what about a not differentiable function as represented in one of the charts ?

Thanks
 
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  • #2
cianfa72 said:
In other words: we know the notion of differentiable function is compatible across charts belonging to the same differential structure, but what about a not differentiable function as represented in one of the charts ?
If it were differentiable in one chart, then it would be in all charts (according to the first part of your sentence). So, if it isn't in one chart, it will not be in the others.
 
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Likes WWGD
  • #3
Sure, thank you!
 

What is a differentiable function?

A differentiable function is a mathematical function that can be smoothly and continuously defined on a given manifold. This means that the function has well-defined derivatives at every point on the manifold, allowing for the calculation of instantaneous rates of change.

What is a manifold?

A manifold is a mathematical space that can be described using coordinates and equations. It is a generalization of the concept of a surface in three-dimensional space, and can have any number of dimensions. Examples of manifolds include curves, surfaces, and higher-dimensional spaces.

How is a differentiable function defined on a manifold?

A differentiable function on a manifold is defined by assigning a value to each point on the manifold. This value can then be used to calculate the function's derivatives at that point. The function must also be continuous and have well-defined derivatives at every point on the manifold.

What is the importance of differentiable functions on manifolds?

Differentiable functions on manifolds have many important applications in mathematics, physics, and engineering. They are used to model real-world phenomena and make predictions about how these phenomena will change over time. They are also crucial in the development of many mathematical theories and techniques.

How do differentiable functions on manifolds differ from regular functions?

Regular functions are defined on a single, fixed set of coordinates. In contrast, differentiable functions on manifolds are defined on a more general space, allowing for greater flexibility and applicability. Additionally, regular functions may not have well-defined derivatives at every point, while differentiable functions on manifolds must have well-defined derivatives at every point.

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