Several Variables Truth

Calculus Level 4

True or False?

For any differentiable function $f(x,y)$ , then $\frac{\partial^2 f(x,y)}{\partial x \partial y}=\frac{\partial^2 f(x,y)}{\partial y \partial x}$

must be satisfied.

True False

1 solution

Hjalmar Orellana Soto
Jan 5, 2017

A counter example is $f(x,y)=x^{y}$ . In general, a function $f(x,y)$ must be at least $C^2$ for $\frac{\partial^2 f}{\partial x\partial y}=\frac{\partial^2 f}{\partial y \partial x}$ to be true, which means the second partial derivatives are continous and differentiable

No. The continuity of second partial derivatives(Schwarz Theorem) is a sufficient but not a necessary condition and differentiability of first order partial derivatives is another sufficient condition(Taylor's Theorem) . Schwarz Theorem and Taylor's Theorem is not true conversely and one of them might be true and the other being untrue. However the example you suggested is perfectly correct.

Arghyadeep Chatterjee - 1 year, 4 months ago

Several Variables Truth

1 solution

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