An Example From Differential Equations

mathimages_10.gif

Clay Ross (U. of the South, Sewanee, Tenn.) suggested the following: f(x, y) = x + x^2/2 - y + y^2/2 + log |x-1| + log |y+1|. The contour-flow image is more informative than a traditional plot. This surface arose from the separable differential equation, y' = x^2 (1 + y) / (y^2 (1-x)). Solutions to the diff. eqn. correspond to contours on the graph. From a geometrical perspective, the surface is interesting in that many of the flows dive into the origin, which is a saddle point.

Spikey Created with Wolfram Mathematica 9.0