The triple product rule in multivariable calculus is widely used. Can a quadruple product rule equation be written for an equation f(x,y,z,z2)=0?
Is there a quadruple product rule?
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multivariable-calculus
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2If you follow the argument given in that wikipedia page, and if I understamnd correctly what you want, you can easily see what the *quadruple product rule* is. – 2012-04-23
2 Answers
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Actually, there is general product rule for $n$-tuple. It is almost the direct consequence of implicit function theorem. Assume $n$ variables satisfies $F(x_1,\ldots,x_n)=0$, we have $\frac{\partial x_i}{\partial x_j}=-{\partial F/\partial x_j\over\partial F/\partial x_i}$ Then multiply all fractions $\frac{\partial x_1}{\partial x_2}\cdots\frac{\partial x_{n-1}}{\partial x_n}\frac{\partial x_n}{\partial x_1}=(-1)^n{\partial F/\partial x_2\over\partial F/\partial x_1}\cdots{\partial F/\partial x_n\over\partial F/\partial x_{n-1}}{\partial F/\partial x_1\over\partial F/\partial x_n}=(-1)^n$
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This is what would make sense to me: $(fghi)'=f'ghi+fg'hi+fgh'i+fghi'$ You can keep adding $n$ functions to this rule.
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7I think you misunderstood what the question is asking about. Look at the link given by anon in his comment. – 2013-11-18