Determining Smoothness Of A Function

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In summary, the conversation discusses the problem of determining whether a given function f defined on a manifold is smooth. The homework equations state that f is smooth if and only if the corresponding function F is smooth, and provide a method for checking this smoothness using partial derivatives. The conversation also explores the relationship between the function and the coordinate chart, and how this affects the calculation of higher derivatives.
  • #1
Alex86
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1. The problem statement:

Given a chart [tex]\varphi[/tex], define a function f by f(p) = x[tex]^{k}[/tex](p), the k-th coordinate of p (where k is fixed). Is f smooth?

2. Homework Equations :

f is smooth (C[tex]^{k}[/tex]) iff F is smooth (C[tex]^{k}[/tex]), where F: [tex]\Re^{n} \rightarrow \Re, F = f \circ \varphi^{-1}[/tex]

The Attempt at a Solution



[tex]\frac{\partial F}{\partial x^{i}} = \sum_{m} \frac{\partial f}{\partial x^{m}} \frac{\partial x^{m}}{\partial x^{i}}[/tex]

[tex]\frac{\partial F}{\partial x^{i}} = \sum_{m} \frac{\partial x^{k}}{\partial x^{m}} \frac{\partial x^{m}}{\partial x^{i}}[/tex]

[tex]\frac{\partial F}{\partial x^{i}} = \delta^{k}_{m} \frac{\partial x^{m}}{\partial x^{i}}[/tex]

[tex]\frac{\partial F}{\partial x^{i}} = \delta^{k}_{i}[/tex]

I'm not sure if this shows what I'm after as I'm not sure exactly what smoothness means in a given situation.

Thanks in advance for any input.
 
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  • #2
I forgot to point out that the function "f(p)" is defined on a manifold where p is a point on the manifold.
 
  • #3
It seems like [tex]f[/tex] can only be defined in the domain of, and relative to, a single coordinate chart; is this what you intend?

Given that, you have the correct calculation. What does the calculation [tex]\partial F/\partial x^i = \delta_{ik}[/tex] tell you about the higher derivatives of [tex]F[/tex]?
 
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  • #4
I'm not entirely sure what you mean by the first question... why would it only be defined in the domain of and relative to a single coordinate chart? Could there not be another chart in an open neighbourhood which also includes p? The chart in the question seems very general to me.

As for the second question does that mean that the function and the chart are [tex]C^{\infty}[/tex] related?
 
  • #5
Alex86 said:
I'm not entirely sure what you mean by the first question... why would it only be defined in the domain of and relative to a single coordinate chart? Could there not be another chart in an open neighbourhood which also includes p? The chart in the question seems very general to me.

Suppose [tex]x: U \to \mathbb{R}^n[/tex] and [tex]y: V \to \mathbb{R}^n[/tex] are two coordinate charts (the same functions you have been denoting [tex]\varphi[/tex]) defined on open neighbhorhoods [tex]U[/tex] and [tex]V[/tex] of [tex]p[/tex]. The [tex]k[/tex]th coordinate functions [tex]x^k: U \to \mathbb{R}[/tex] and [tex]y^k: V \to \mathbb{R}[/tex] need not have anything to do with each other.

Alex86 said:
As for the second question does that mean that the function and the chart are [tex]C^{\infty}[/tex] related?

No, that's not what I meant. Denoting the chart again by [tex]x[/tex] instead of [tex]\varphi[/tex], [tex]F = x^k \circ x^{-1} : x(U) \to \mathbb{R}[/tex] is a function on an open subset [tex]x(U)[/tex] of [tex]\mathbb{R}^n[/tex]. You know its partial derivatives: [tex]\partial F/\partial x^i = \delta_{ik}[/tex]. This contains all the information you need to compute all the higher derivatives of [tex]F[/tex], and thus tell whether [tex]F[/tex] is a [tex]C^\infty[/tex] function (from an open subset of [tex]\mathbb{R}^n[/tex] to [tex]\mathbb{R}[/tex]; no need for coordinate charts here).
 

FAQ: Determining Smoothness Of A Function

What is the definition of smoothness in a function?

Smoothness in a function refers to its level of continuity and differentiability. A smooth function is one that has no abrupt changes or breaks in its graph and is differentiable at every point. In other words, a smooth function is one that has no sharp corners, cusp points, or vertical tangents.

How is the smoothness of a function determined?

The smoothness of a function can be determined by looking at its graph and analyzing its behavior. A function is considered smooth if it has a continuous and differentiable graph. This means that there are no sudden jumps or disruptions in the graph, and the slope of the graph is defined at every point. Additionally, the higher the number of continuous derivatives a function has, the smoother it is considered to be.

Can a function be both smooth and discontinuous?

No, a function cannot be both smooth and discontinuous. A discontinuous function is one that has a break or jump in its graph, which means it is not continuous. Smoothness requires continuity, so a function cannot be smooth if it is discontinuous.

What are some common methods for determining the smoothness of a function?

There are several methods for determining the smoothness of a function, including analyzing the behavior of its graph, calculating its derivatives, and using mathematical theorems such as the Mean Value Theorem or Rolle's Theorem. In some cases, computer software or algorithms can also be used to determine the smoothness of a function.

Why is it important to determine the smoothness of a function?

Determining the smoothness of a function is important because it helps us understand the behavior of the function and its graph. A smooth function is easier to work with and analyze, and it can also provide insights into the nature of the relationship between its input and output. Additionally, knowing the smoothness of a function can help in solving problems and making predictions in various fields such as physics, engineering, and economics.

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