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NickMusicMan
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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 [itex]\partial[/itex][itex]\alpha[/itex]f to refer to "the generic partial derivative of order [itex]|\alpha|[/itex], since the order of differentiation doesn't matter". ([itex]\alpha[/itex] 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 :)
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 [itex]\partial[/itex][itex]\alpha[/itex]f to refer to "the generic partial derivative of order [itex]|\alpha|[/itex], since the order of differentiation doesn't matter". ([itex]\alpha[/itex] 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 :)