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**1. The problem statement, all variables and given/known data**

[itex]f(x) = f(x_1,...,x_N) : R^N \mapsto R[/itex] has continuous partial derivatives. Assume that for a point a in [itex] R^N , \frac{\partial f}{\partial x_i}(a) \neq 0[/itex] for all i = 1...N.

The implicit function theorem says that near a, the equation [itex] f(x) = f(a)[/itex] can be used to express each [itex] x_i[/itex] in terms of other [itex] x_j s[/itex]. Prove the following equality at a:

**2. Relevant equations**

[tex] (-1)^N = \frac{\partial x_1}{\partial x_2} \cdot \frac{\partial x_2}{\partial x_3} ... \cdot \frac{\partial x_N-1}{\partial x_N} \cdot \frac{\partial x_N}{\partial x_1} [/tex]

[itex] \frac{\partial}{\partial y_j} f(g(y)) = \nabla f(g(y)) \cdot \frac{\partial}{\partial y_j} g(y) [/itex] (chain rule)

**3. The attempt at a solution**

I've tried writing something like [itex] x_1 = x_1(x_2,...,x_N) [/itex] to make each one a function of the other and tried using the chain rule on that, but it gets really messy quickly. I'm really unsure about the significance of the implicit function theorem or where to start with this problem. I found a similar law called the triple product rule which seems to be this equality in N = 3, but i've been unable to extend it to general N. Any help would be greatly appreciated.

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