The animation below illustrates a theorem in topology: if you are on an island and count the number of peaks or sinks and subtract the number of mountain passes, we always get one. For example, there could be one mountain or two mountains and one mountain pass. This theorem assumes that for all the critical points the discriminant D is nonzero. In the animation, the color at every point is the curvature K which is defined as D/(1+|f'|^2) where f' is the gradient of f and D=f_{xx} f_{yy} - f_{xy}^{2} is the discriminant (which is defined also at points which are not critical points). You see that the curvature at the critical point agrees with the discriminant D which appears in the second derivative test. |