# Math 22a Fall 2018

## 22a Linear Algebra and Vector Analysis

# Clairaut counter example

We see here an illustration of Clairaut's theorem first for the function which is given in polar coordinates asg(r,t) = r^{2}sin(4t)

f(r,t) = r^{2}sin(2t)

Clairaut theorem: f in C^{2} implies f_{xy}=f_{yx}. |
Clairaut counter example: there is g in C^{1} for which Clairaut fails |

_{xy}and g

_{yx}are different. The function g

_{x}gives the slope in the x direction and g

_{xy}is the rate of change of that slope. The function g

_{y}gives the slope in the y direction and g

_{yx}gives the rate of change of that slope. You see that unlike for the function f, where the two rate of changes agree, in the second example for g, the rate of change has different sign.

f_{xy} | f_{yx} |

g_{xy} | g_{yx} |