# 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: $13.61$