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Definition confusion:

I wish to show that $$f(x,y)={-y\over x}$$ and $$g(x,y)=\log |x|$$ are functionally independent on some domain.

What does that mean? What do I have to show? And how does one choose the domain?

Thank you.


This is related to question 2 on P. 84 in this book. In particular, the note in the square brackets. However, I don't know what exactly that is and why we would like to do that.

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    Why do you wish to show it?2012-11-24
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    @JonasMeyer: This is related to question 2 on P. 84 in [this book](http://www.scribd.com/doc/75041721/Maciej-Dunajski-Solitons-Instantons-and-Twistors). In particular, the note in the square brackets. However, I don't know what exactly that is and why we would like to do that.2012-11-24

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It means that the gradients $\nabla{f},\nabla{g}$ are linearly independed in that domain. i.e. the Jacobian $J_F$ of the function $F(x,y)=(f(x,y),g(x,y))$ has full rank.
So calculate $J_F$ and find the domain such that $\operatorname{rank}{(J_f)}=2 $.

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    What about when $F(x,y,z) = (f(x,y,z),g(x,y,z))$? The Jacobian is then 2-by-3. Thanks!2013-07-06
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    In that case we want the Jacobian to have rank $2$. In any case we want the Jacobian to have rank equal to the number of the functions.2013-07-07
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    @P.. "_we want the Jacobian to have rank equal to the number of the functions_" Why does that mean the functions are functionally independent?2018-01-11
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    @Geremia: See https://math.stackexchange.com/questions/1110763/intuition-behind-functional-dependence2018-01-11
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Well, in the book Holomorphic Morse Inequalities and Bergmann Kernels(p.103), I think it means that if there are $a,b\in\mathbb R$, such that $$af+bg=0,$$ we have $a=b=0$.