1

Are the following 2 operations guaranteed to give the same result?

$\frac{\partial}{\partial x}(\frac{\partial}{\partial y}(f(x, y)))$

$\frac{\partial}{\partial y}(\frac{\partial}{\partial x}(f(x, y)))$

MrChickenNuggetBadman
  • 61
  • 1
    Not if $f$ is discontinuous or if $f_{xy} $ and $f_{yx}$ are not continuous. – cpiegore Feb 02 '23 at 20:29
  • 1
    Assuming $f$ is defined on $\mathbb{R}^2$ and $f{\in}{C}^2(\mathbb{R}^2)$, yes. It's a consequence of Schwarz's theorem. – simo210 Feb 02 '23 at 20:30

0 Answers0