Equality of mixed partial derivatives of order >2

[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 C^k function, but I am having trouble figuring out exactly how this works.

For example, take f:ℝ² → ℝ .

f_xxy=f_xyy is not true, right?


Folland's Advanced Calculus says that in Taylor's theorem we can use the notation \partial^\alphaf to refer to "the generic partial derivative of order |\alpha|, since the order of differentiation doesn't matter". (\alpha is a multi-index)

But this doesn't 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?

Thanks in advance for your replies :)

 

[HallsofIvy]

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 C^k function, but I am having trouble figuring out exactly how this works.

For example, take f:ℝ² → ℝ .

f_xxy=f_xyy is not true, right?

Yes, that's right- but not really what you are asking. After all, f_xx is not the samer as f_yy. What f_xy= f_yx says is that, as long as the derivatives are continuous, the order of the derivatives does not matter.

Folland's Advanced Calculus says that in Taylor's theorem we can use the notation \partial^\alphaf to refer to "the generic partial derivative of order |\alpha|, since the order of differentiation doesn't matter". (\alpha is a multi-index)

But this doesn't 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?

Thanks in advance for your replies :)

Apparently you are missing the fact that "order" does not mean changing the number of derivatives with respect to the different variables. "yxx" and "xyx" are different orders of the same thing, "yxx" and "xyy" are NOT.