Question

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

Answer 1

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

Answer 2

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 ;)

Answer 3

That is unambiguously terrifying...

Answer 4

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

Answer 5

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

Answer 6

just do it in 2 variables and it already becomes unintuitable

Answer 7

@ganeshie8