# Functional independence

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? –  Jonas Meyer Nov 24 '12 at 3:11
@JonasMeyer: 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. –  George Nov 24 '12 at 3:35

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 $\operatorname{rank}{(J_f)}=2$.
What about when $F(x,y,z) = (f(x,y,z),g(x,y,z))$? The Jacobian is then 2-by-3. Thanks! –  WishingFish Jul 6 '13 at 0:24
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. –  P.. Jul 7 '13 at 8:23