Second Derivative Pathology

thingiverse

This is a model of the graph of a function f(x,y) where two mixed partial derivatives, f_xy and f_yx, both exist but are unequal. Moving little indicators along strings reveals that one mixed partial is positive while the other is negative. As a bonus, rotating the graph of f(x,y) by an angle of pi/8 gives a new function g(x,y), which has the property that its second derivative test for local extrema yields incorrect results. The formula for f(x,y) is given as f(x,y) = r^2 * sin(4*theta) = 4*x*y*(x^2 - y^2) / (x^2 + y^2).