# Equality of mixed partial derivatives of order >2

1. Jan 3, 2012

### NickMusicMan

I know that for any C2 function, the mixed second-order partials are equal, and I see that this should extend inductively to a statement about the kth partials of a Ck function, but I am having trouble figuring out exactly how this works.

For example, take f:ℝ2 → ℝ .

fxxy=fxyy is not true, right?

Folland's Advanced Calculus says that in Taylor's theorem we can use the notation $\partial$$\alpha$f to refer to "the generic partial derivative of order $|\alpha|$, since the order of differentiation doesnt matter". ($\alpha$ is a multi-index)

But this doesnt make sense to me, considering that there are multiple possibilities for the order>2 partials, even when restricting it to mixed partials. Is this an oversight in Folland's wording, or am I missing something?