How to do multivariable Taylor expansion? Are there any formula for that?

Question

How to do multivariable Taylor expansion?  Are there any formula for that?

reemii · Answer

https://en.wikipedia.org/wiki/Taylor_series#Taylor_series_in_several_variables

reemii · Answer

Basically you must, for the k-th term, compute all possible k-th derivatives: $\frac{\partial^k}{\partial{x_1^k}}f, \frac{\partial^{k-1}}{\partial{x_1^{k-1}}} \frac{\partial}{\partial{x_2}}f, \dots$ it's okay to stop at the second term ;)

thomas5267 · Answer

That is unambiguously terrifying...

thomas5267 · Answer

Just for the sake of being pathological, what if some partial derivatives don't commute?  Does the Taylor series even exists?

reemii · Answer

No idea. But this is used in statistics (to 1st or 2nd order at most).

irishboy123 · Answer

just do it in 2 variables and it already becomes unintuitable

thomas5267 · Answer

@ganeshie8