Is f_m Smooth When f Is a Smooth Map Between Manifolds?

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In summary, smooth maps are functions between differentiable manifolds that maintain smoothness and can be continuously differentiated. They are a subset of regular maps, with the added requirement of continuous derivatives. Smooth maps are important in mathematics, particularly in differential geometry, and have practical applications in modeling and analyzing complex systems. A diffeomorphism is a special type of smooth map that preserves the geometry and topology of a manifold, while a regular smooth map may distort these properties.
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slevvio
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Hello everyone, I just had a quick question I was hoping somebody could answer.

If [itex]f: M \times N \rightarrow P[/itex] is a smooth map, where [itex]M,N[/itex] and [itex]P[/itex] are smooth manifolds, then is it true for fixed [itex]m[/itex] that [itex]f_m : N \rightarrow P[/itex] is smooth, where [itex]f_m (n) = f(m,n)[/itex]?

Any help would be appreciated.
 
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  • #2
Of course, since the equivalent statement when M=R^m, N=R^n is true.
 
  • #3
ok thank you!
 

FAQ: Is f_m Smooth When f Is a Smooth Map Between Manifolds?

What are smooth maps?

Smooth maps are functions between differentiable manifolds that preserve smoothness. In other words, they are maps that can be continuously differentiated.

How are smooth maps different from regular maps?

Smooth maps are a subset of regular maps, which are simply functions between sets. What sets smooth maps apart is that they are required to have continuous derivatives, while regular maps may not have this requirement.

What is the importance of smooth maps in mathematics?

Smooth maps are essential in differential geometry, which is a branch of mathematics that studies smooth manifolds. They also have many applications in physics, engineering, and other fields that involve studying continuous systems.

How are smooth maps used in practical applications?

Smooth maps are used in practical applications to model and analyze complex systems. For example, they can be used to describe the motion of objects, the flow of fluids, or the behavior of electrical circuits. They are also used in computer graphics to create smooth and realistic animations.

What is the difference between a smooth map and a diffeomorphism?

A diffeomorphism is a special type of smooth map that is both one-to-one and onto, with a smooth inverse function. This means that a diffeomorphism preserves all of the geometry and topology of a manifold. In contrast, a smooth map may not have these properties and can distort the geometry and topology of a manifold.

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