Determining Smoothness Of A Function

[Alex86]

1. The problem statement:

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

2. Homework Equations :

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

The Attempt at a Solution

$$\frac{\partial F}{\partial x^{i}} = \sum_{m} \frac{\partial f}{\partial x^{m}} \frac{\partial x^{m}}{\partial x^{i}}$$

$$\frac{\partial F}{\partial x^{i}} = \sum_{m} \frac{\partial x^{k}}{\partial x^{m}} \frac{\partial x^{m}}{\partial x^{i}}$$

$$\frac{\partial F}{\partial x^{i}} = \delta^{k}_{m} \frac{\partial x^{m}}{\partial x^{i}}$$

$$\frac{\partial F}{\partial x^{i}} = \delta^{k}_{i}$$

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.

 

[Alex86]

I forgot to point out that the function "f(p)" is defined on a manifold where p is a point on the manifold.

 

[ystael]

It seems like $$f$$ 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 $$\partial F/\partial x^i = \delta_{ik}$$ tell you about the higher derivatives of $$F$$?

 

[Alex86]

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 $$C^{\infty}$$ related?

 

[ystael]

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

As for the second question does that mean that the function and the chart are $$C^{\infty}$$ related?

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