Second Derivative Pathology

thingiverse

This is a model of the graph of a function f(x,y) for which the two mixed partial derivatives f_xy and f_yx both exist but are unequal. By pushing the little indicators along the strings, one can see that one of the mixed partials is positive, while the other is negative. As a bonus, if the graph of f(x,y) is rotated by an angle of pi/8 to give a new function g(x,y), then g(x,y) has the property that the "second derivative test" for local extrema gives the wrong answer. The formula for f(x,y) is given by f(x,y) = r^2 * sin(4*theta) = 4*x*y*(x^2 - y^2) / x^2+y^2