Ron Goetz and I have developed a technique and Mathematica package (called ContoursAndFlows) to show the surface that is the graph of z = f(x, y) by setting up a coordinate system based on contours and the curves orthogonal to them (flows: the paths a drop of water would take under gravity). This allows us to get very good images of graphs that are difficult, or impossible, to see using standard Cartesian grids. (References: Our two articles in Mathematica in Education and Research) Here are several examples.
A Double Torus, Oriented Vertically
Unequal Mixed Partial Derivatives
Discontinuous, But Partial Derivatives Exist
Failure of the Only-Critical-Point-in-Town Test
An Example From Differential Equations
Rotationally Symmetric Venn Diagrams
Cutting a Solid Torus into 13 Pieces
| Created by Wolfram Mathematica 6.0 (28 October 2007) | |