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Thread: Proving: Partial Derivatives

  1. #1
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    Proving: Partial Derivatives

    Assume that $\displaystyle f_{xy}$ and $\displaystyle f_{yx}$ are continuous and that $\displaystyle f_{yxx}$ exists. Show that $\displaystyle f_{xyx}$ also exists and that $\displaystyle f_{yxx} = f_{xyx}$.
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  2. #2
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    $\displaystyle f_{yxx}$ means $\displaystyle (f_{yx})_x$. Since $\displaystyle f_{xy}$ and $\displaystyle f_{yx}$ are continuous, $\displaystyle f_{yx}= f_{xy}$.
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