Breaking Multivariable Calculus

A series of interactive 3D prints for understanding pathological multivariable functions.

I made a series of interactive 3D printed models designed to illustrate how important theorems and facts in multivariable calculus break down when the underlying assumptions are violated. The examples themselves are not new, but the ability to interact with a physical model is meant to help learners understand what is happening. The hope is that discussion of these pathological functions can serve as a bridge between studying multivariable functions in a calculus course and a real analysis course. See my article in Mathematics Magazine.

A non-differentiable function, all of whose directional derivatives are defined.

A function with no limit at the origin

A function that is not twice differentiable.