Question

Use Clairaut’s Theorem to show that if the third-order derivatives of \(f\) are continuous, then \({f_{xyy}} = {f_{yxy}} = {f_{yyx}}\).

Short answer

A partial differential equation (PDE) is a mathematical equation that establishes relationships between the partial derivatives of a multivariate function.

Step-by-step solution

Given Data,

It is given that the third order partial derivatives are continuous. This implies that the second order derivatives\({f_{xy}}\)and \({f_{yx}}\) are also continuous.

Then by Clairaut’s theorem,

\({f_{xy}} = {f_{yx}}\) …… (1)

Differentiate with respect to \(y\).

\(\frac{\partial }{{\partial y}}\left( {{f_{xy}}} \right) = \frac{\partial }{{\partial y}}\left( {{f_{yx}}} \right)\)

\({\rm{ }}{f_{yxy}} = {f_{yyx}}\)…… (2)

\(\begin{aligned}{c}{f_{yxy}} = {\left( {{f_{yx}}} \right)_y}\\ = {\left( {{f_{xy}}} \right)_y}\end{aligned}\)

\({\rm{ }} = \left( {{f_{xyy}}} \right)\)…… (3)

Hence equation (2) and (3) combine to give

\({f_{xyy}} = {f_{yxy}} = {f_{yyx}}\).