Clairaut’s Theorem

Conditions

All mixed partial derivatives are equal

$$\frac{\partial }{\partial x_i}\left(\frac{\partial f}{\partial x_j}\right)=\frac{\partial }{\partial x_j}\left(\frac{\partial f}{\partial x_i}\right)$$ $$\begin{gathered} \text{2D}: \\ {f}_{xy}=f_{yx} \\ \text{3D}: \\ {f}_{xy}=f_{yx} \\ {f}_{yz}=f_{zy} \\ {f}_{xz}=f_{zx} \end{gathered}$$

Theorem

• Mixed partials are continuous at $P_0$ → Clairaut’s conditions are satisfied at $P_0$
• Useful in cases where we know any partial derivative is going to be continuous
  • Such as
    • Polynomials
    • Rational (where denominator $\ne$ 0)
    • $f=g(t)$ where $g^{\prime }(t)$ is continuous
• Extensions
  • If all partial derivatives of order n are continuous
  • Then the order of taking partial derivatives doesn’t matter (they are commutative)