3
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ What is the triple product rule? $\endgroup$ – Qiaochu Yuan Apr 23 '12 at 3:46
  • 3
    $\begingroup$ @Qiaochu: en.wikipedia.org/wiki/Triple_product_rule $\endgroup$ – anon Apr 23 '12 at 3:47
  • 2
    $\begingroup$ If 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. $\endgroup$ – Mariano Suárez-Álvarez Apr 23 '12 at 3:53
2
$\begingroup$

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 $$\frac{\partial x_1}{\partial x_2}\cdots\frac{\partial x_{n-1}}{\partial x_n}\frac{\partial x_n}{\partial x_1}=(-1)^n{\partial F/\partial x_2\over\partial F/\partial x_1}\cdots{\partial F/\partial x_n\over\partial F/\partial x_{n-1}}{\partial F/\partial x_1\over\partial F/\partial x_n}=(-1)^n$$

$\endgroup$
-1
$\begingroup$

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.

$\endgroup$
  • 7
    $\begingroup$ I think you misunderstood what the question is asking about. Look at the link given by anon in his comment. $\endgroup$ – Willie Wong Nov 18 '13 at 13:40

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.