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I am doing some self-study in math. The problem below is from a Calculus text book.

Problem:

Establish the fact, widely used in hydrodynamics, that if $f(x,y,z) = 0$, then \begin{eqnarray*} \Big(\frac{\partial x}{\partial y}\Big)_z \Big(\frac{\partial y}{\partial z}\Big)_x \Big(\frac{\partial z}{\partial x}\Big)_y &=& 0 \\ \end{eqnarray*} ( Hint: Express all the derivatives in terms of the formal partial derivatives $\frac{\partial f} {\partial x}$, $\frac{\partial f}{\partial y}$ and $\frac{\partial f}{\partial z}$. )

When I see this question, I interpreter it to mean that we are asked to prove the above fact for any function. So, if I choose the particular function $f(x,y,z) = 0$ then the products of its partial derivatives should be -1. However, in this case, since all the partial derivatives are $0$, their product is $0$. What am I missing?

Bob

Bob
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    Should the last term in the product be $(\partial z / \partial x)_y$? – Robert Lewis Jul 06 '17 at 16:33
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    @RobertLewis Yes it should be. I will fix the post. Thank you very much. – Bob Jul 06 '17 at 16:36
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    Glad to help out. – Robert Lewis Jul 06 '17 at 16:38
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    Are you sure??? Triple product rule – Violapterin Jul 06 '17 at 17:03
  • @Aminopterin I am not sure. The way I see it, the link you sent me, which was very helpful, tends to imply that the partial derivatives have to be different from zero on the integral. Is that right? (I suspect it is not right). If it is then it seems to me the question should be changed? Bob – Bob Jul 06 '17 at 18:18
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    Yes, you need the partials to be nonzero in order to use the implicit function theorem to say that each variable is a function of the other two. See here: https://math.stackexchange.com/questions/942457/understanding-frac-partial-x-partial-y-frac-partial-y-partial-z-frac – Hans Lundmark Jul 06 '17 at 20:42

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