Clairaut's Theorem

Definition

Clairaut’s Theorem (also known as Schwarz’s Theorem) states that for a function of several variables $f(x, y)$, if the mixed partial derivatives $f_{xy}$ and $f_{yx}$ are continuous on an open disk $D$, then they are equal at every point in $D$: $f_{xy}(a, b) = f_{yx}(a, b)$

• How to read: “The mixed partial derivative f x y at a, b equals f y x at a, b.”
• Meaning: On a smooth surface, the order of differentiation does not matter for mixed partials—$\partial^2 f / \partial y \partial x = \partial^2 f / \partial x \partial y$.

Why It Matters

It simplifies complex multivariable problems by proving that the order of differentiation is irrelevant for smooth surfaces, ensuring mathematical consistency in physics and engineering.

Core Concepts

• Mixed Partials: Refers to differentiating a function with respect to one variable and then another (e.g., $\frac{\partial^2f}{\partial y \partial x}$).
  • How to read: “The second partial derivative of f with respect to y and x.”
  • Meaning: Differentiate first with respect to $x$, then $y$ (read right-to-left in subscript order).
• Requirement of Continuity: The theorem only holds if the second-order partial derivatives are continuous. If they are discontinuous, the order of differentiation may matter.
• Higher Orders: The theorem extends to higher-order partial derivatives (e.g., $f_{xxy} = f_{xyx} = f_{yxx}$) provided the continuity conditions are met.
  • How to read: “The mixed partial derivative f x x y equals f x y x, which also equals f y x x.”
  • Meaning: All permutations of three partial differentiations agree when the relevant partials are continuous.

Connected Concepts

• Partial Derivatives
• Second-Order Partial Derivatives
• Total Differential
• Conservative Vector Fields
• Symmetry
• Path Independence Test
• First Principles Thinking