(adsbygoogle = window.adsbygoogle || []).push({}); 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. Relevant 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]

3. 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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# Homework Help: Determining Smoothness Of A Function

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