Question

How many mixed partial derivatives of order 3 can a function of 3 variables have and why?

Answer 1

It depends on the nature of the function, if it's 1st order continuous we have 3*2*1=6 for the purely mixed partials (all 3 variables are different) and 2*3=6 from when we differentiate twice with respect to one variable and once with respect to any other variable, lasty we have 3 more for the non-mixed derivatives - in total 6+6+3=15 partial derivatives.
But if it's 2nd or 3rd order continuous the order of the 2nd and 3rd mixed partials doesn't matter, respectively, so we get fewer.

Answer 2

Hmm, counted them wrong, there must simply be 3*3*3=27 if it's 1st order continuous.

Answer 3

Since there are 3 permutations for the 6 twice/once partials =18, plus 6 from pure mixed and 3 from non-mixed=27

Answer 4

So 1st order continuous there are 27.

The case for 3rd order continuous is also easy, since the order of the mixed partials doesn't matter, so we get 1 from pure mixed, 6 from differentiating twice with respect to one variable and once with respect to another and 3 from the pure partials. So we get 10 for 3rd order continuous functions.

Answer 5

Thank you, I understand it now.