Clairaut's Theorem

Theorem

Let $\map f {x, y}$ be a function of the two independent variables $x$ and $y$.

Let $\map f {x, y}$ be continuous.

Let the partial deriviatives of $f$ also be continuous.


Then:

$\dfrac {\partial^2 f} {\partial x \partial y} = \dfrac {\partial^2 f} {\partial y \partial x}$


Proof


This theorem requires a proof.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.


Also known as

is also known as Schwarz theorem, for Karl Hermann Amandus Schwarz.


Source of Name

This entry was named for Alexis Claude Clairaut.


Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 13$: Derivatives: Partial Derivatives: $13.61$
  • 2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 15$: Derivatives: Partial Derivatives: $15.61.$
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.