Name of partial derivatives where the order of differentiation can be reversed.

It is known as Clairaut's theorem.

Suppose F is defined on a disk D that contains the point (a,b). If the functions $$\frac{\partial{F}}{\partial{x}\partial{y}}$$ and $$\frac{\partial{F}}{\partial{y}\partial{x}} $$ are both continuous on D, then $$\frac{\partial{F}}{\partial{x}\partial{y}}(a,b)=\frac{\partial{F}}{\partial{y}\partial{x}}(a,b)$$


I suspect that the equality $$\frac{\partial{F}}{\partial{x}\partial{y}}=\frac{\partial{F}}{\partial{y}\partial{x}},$$ interpreted as a pde, have no special name because it is not commonly interpreted as a pde.

On the other hand, understood as "a condition on" or "a property of" a certain function $F$, the said equality is usually referred to as

  • Equality of mixed partial derivatives.
  • Symmetry of second derivatives.
  • Symmetry of the hessian matrix.

Tags:

Terminology

Multivariable Calculus

Partial Differential Equations

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