# Is There a Faster Way to Prove Smoothness on Manifolds?

• I
• JonnyG
In summary, a function ##F: M \rightarrow N## between smooth manifolds is considered smooth if, for any point ##p## on ##M##, there exist smooth charts ##(U, \phi)## containing ##p## and ##(V, \psi)## containing ##F(p)##, such that the composition ##\psi \circ F \circ \phi^{-1}## is a smooth map from ##\phi(U \cap F^{-1}(V))## to ##\mathbb{R}^n##. This can be used to show that a function is smooth without explicitly constructing a bijection between manifolds and verifying the smoothness of its coordinate representation. The inverse function theorem
JonnyG
So if ##M, N## are smooth manifolds then ##F: M \rightarrow N## is smooth if given ##p \in M##, there is a smooth chart ##(U, \phi)## containing ##p## and a smooth chart ##(V, \psi)## containing ##F(p)## such that ##\psi \circ F \circ \phi^{-1}: \phi(U \cap F^{-1}(V)) \rightarrow \mathbb{R}^n## is smooth.

If I wanted to prove that a given function was smooth, are there any faster ways other than showing that its coordinate representation is smooth? For example, I just did a question where I had to show that ##T(M \times N)## is diffeomorphic to ##T(M) \times T(N)##. I had to explicitly construct a bijection between the two manifolds then show that the coordinate representations of ##F## and ##F^{-1}## were smooth. This was a big pain. I wish there was a theorem I could have appealed to instead.

Inverse function theorem?
Sections of the tangent bundle?

It really depends on what you have seen already.

Thanks, micromass. I haven't learned the inverse function theorem on manifolds yet, but I suppose it's the usual inverse function theorem applied to the coordinate representation of the map I'm interested in. I am still early in my study of smooth manifolds - I'll be more patient.

## 1. What is the definition of smoothness in terms of mathematics?

Smoothness is a mathematical concept that refers to the property of a function being continuously differentiable. This means that the function has derivatives of all orders, meaning it is infinitely differentiable.

## 2. How is smoothness proven in mathematics?

In mathematics, smoothness is proven by showing that a function has continuous derivatives of all orders. This can be done through various methods, such as using the definition of differentiability or using the Mean Value Theorem.

## 3. What is a simpler way to prove smoothness?

A simpler way to prove smoothness is by using the concept of analytic functions. An analytic function is infinitely differentiable and can be expressed as a power series. Therefore, if a function can be represented as a power series, it is automatically proven to be smooth.

## 4. Can smoothness be proven for all types of functions?

No, smoothness can only be proven for functions that are defined on a continuous domain and have continuous derivatives of all orders. Functions that are not continuous or have discontinuous derivatives cannot be proven to be smooth.

## 5. Why is proving smoothness important in mathematics?

Proving smoothness is important in mathematics because it allows us to understand the behavior of a function and make predictions about its values. It also helps us to solve problems and equations involving the function, as well as analyze its rate of change and curvature.

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