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

A Double Torus, Oriented Vertically

A Fubini Example

A Flattened Sphere

Straight and Parabolic Limits

Unequal Mixed Partial Derivatives

Discontinuous, But Partial Derivatives Exist

Failure of the Only-Critical-Point-in-Town Test

An Illustrative Example

An Example From Differential Equations

Rotationally Symmetric Venn Diagrams

Cutting a Solid Torus into 13 Pieces

Created with Wolfram Mathematica 9.0