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bigli
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can you be given a suitable smooth atlas to the subset M of plane that M to be a differentiable manifold? M={(x,y);y=absolute value of (x)}
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bigli said:How am I use open sets for (0,0) of M?
bigli said:problem is to find a suitable function that must to be differentiable itself and its inverse.but how function and its inverse to be differentiable at (0,0)?
A smooth atlas of a differentiable manifold is a collection of charts or coordinate systems that covers the entire manifold and allows for smooth transitions between different charts.
A smooth atlas is specifically designed for differentiable manifolds, meaning that the transition maps between charts are smooth (infinitely differentiable). In a regular atlas, the transition maps may not necessarily be smooth.
A smooth atlas allows for a consistent and well-defined notion of differentiability on the manifold. This is necessary for studying the properties and behaviors of differentiable functions on the manifold.
A smooth atlas is constructed by choosing a set of charts that cover the manifold, and ensuring that the transition maps between charts are smooth. This can be done by hand or using more advanced mathematical techniques.
Yes, a smooth atlas can have an infinite number of charts, as long as they cover the entire manifold and the transition maps between any two charts are smooth. This is often the case for more complex differentiable manifolds.