Most interesting exercises about the implicit and inverse function theorems

Question

I am a TA in a multivariable calculus course this semester. Right now I am writing the exercise session which deals with the implicit function theorem, inverse function theorem and open function theorem (i.e. submersions are open maps).

I am looking for your favorite/most interesting exercises which I can use/adapt. I especially like exercises which combine application of the theorems with some geometrical problem, but I am open to anything interesting. For instance, I give one exercise which deals with computing the tangent to the intersection curve of two surfaces (which requires computing the derivatives implicitly).

Thanks in advance!

Answer 1

I find the geometry of the helicoid to be pretty interesting because it illustrates in concrete visual terms many advanced topics that are often difficult for students to grasp on first exposure. And the ImFT is the perfect tool for exploring some of its features.

1. The implicit equation for the helicoid, written as $F(x,y,z)= x\sin z- y \cos z =0$ defines a spiral staircase surface that is smooth (everywhere that grad $F$ is nonzero on the surface). One can use the ImFT to investigate where the projection onto the $(x,y)$ plane is locally smoothly invertible. This exercise reveals that the helicoid is an example of what is technically known as an infinitely-sheeted regular covering surface over the punctured plane $(x,y)\ne (0,0)$.

It is also fun to use computer graphics to explore other level sets of the form $F(x,y,z)=x\sin z- y \cos z =C$ when $C\ne 0$.

2. As another exercise, one can calculate $<z_x, z_y>$ implicitly and verify that it is the vector field $\frac{ <-y,x>}{x^2+y^2}$. This is interesting since it prepares the way for a subsequent discussion of why this planar vector field is locally conservative but not globally conservative. (The potential function $z= arctan (y/x)$ is globally multiple-valued but locally single-valued.)

3. One can also use this surface to illustrate geometrically the concept of winding number of a closed loop in the punctured plane, and give a heuristic explanation for why this winding number is invariant under deformations of the base path. (The amount of work we do climbing up the helicoid is not affected by small perturbations of the path.)

4. More generally, one can explore the phenomena of caustics in the plane by looking at the exceptional places where the ImFT fails. Caustics are visually captivating topics with interesting connections to many physical applications eg. ray optics. The simplest examples of caustics formed by rays are of the general form $ F(x,y,z)=a(z) x+ b(z) y - c(z)=0$ which is a parametrized family of lines in the $(x,y)$ plane whose coefficients can be nonlinear functions of the parameter $z$. Regarded as an implicitly defined surface $S$ in $(x,y,z)$ space, this family of lines comes by projecting $S$ onto the $(x,y)$ plane. Consider the points on the surface $S$ where the normal vector $\nabla F$ to the surface is either zero or points horizontally, i.e. $F_z=0$. This is where the projection map may fail to be locally smoothly invertible. These bad points determine the caustic which is also called the envelope of the family of lines.