What are the conditions that a function $f(x,y)$ should satisfy for the partial derivatives $f_{xy}$ and $f_{yx}$ to be equal?
When do partial derivatives fail to commute?
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derivatives
continuity
partial-derivative
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1You need the function to be twice continuously differentiable (aka of class $C^2$). See https://en.wikipedia.org/wiki/Symmetry_of_second_derivatives#Schwarz.27s_theorem – 2017-02-02
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0@Crostul since the OP commented on my post I came back to this question and just saw your comment. Note that the twice differentiability is **sufficient** but not **necessary**, so you don't "need" this. – 2017-03-26
1 Answers
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Clairaut’s Theorem
Suppose that $f$ is defined on a disk $D$ that contains the point $(a,b)$. If the functions $f_{xy}$ and $f_{yx}$ are continuous on this disk then $$f_{xy}(a,b) =f_{yx}(a,b)$$
We can actually restrict ourselves a bit less and let $D$ be any open subset of $\mathbb{R}^2$, which generalizes nicely to
Suppose $f$ is a function of variables defined on an open subset $D$ or $\mathbb{R}^n$. Suppose all mixed partials with each possible number of and combination of differentiations in each input variable exist and are continuous on $D$. Then, all the mixed partials are continuous.
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0Can you give a simple example of a function $f(x,y)$ where $f_{xy}\neq f_{yx}$? @Brevan Ellefsen – 2017-03-26
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0@SRS see here https://calculus.subwiki.org/wiki/Failure_of_Clairaut's_theorem_where_both_mixed_partials_are_defined_but_not_equal – 2017-03-26