Clairaut's Theorem
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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
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Also known as
- Clairaut's Theorem 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): $\S 13$: Partial Derivatives: $13.61$