The Theorem That Unites Different Kinds of Calculus

March 14th, 2019

Authored by Evelyn Lamb

On this episode of our podcast My Favorite Theorem, my cohost Kevin Knudson and I got to talk with Robert Ghrist. Dr. Ghrist has a joint professorship in the departments of mathematics and of electrical and systems engineering at the University of Pennsylvania. You can listen to the episode here or a kpknudson.com, where there is also a transcript.

Dr. Ghrist chose a theorem (one that doesn’t have a name of its own, so we’ll just have to call it Ghrist’s favorite theorem) that to him sums up a deep relationship between discrete analogs of calculus. Calculus is the study of continuous change over time, but many important systems change discretely. (For example, think of the birth of a new organism or generation of organisms.) Some tools from calculus can be used to study these systems as well, and Ghrist’s favorite theorem is one way to form a link between the two subjects.

In continuous calculus, the derivative is a central object of study. The derivative is a measure of change in a system over time, so the discrete analog of the derivative is a shift. The shift operator acts on a function by taking an input to the output of the function at the next time step. If we call the shift operator E, as Ghrist does, E(f(x))=f(x+1). Ghrist’s favorite theorem connects the shift operator to a procedure in continuous calculus of exponentiating a function, that is, of taking e to the power of a function.

While the idea of raising a number to a functional power doesn’t immediately seem meaningful, the magic of Taylor’s theorem allows us to define the process into something meaningful. In fact, Taylor’s theorem is the best way to make sense of raising e to irrational or imaginary powers. Taylor’s theorem allows us to approximate more difficult functions with polynomials. (These polynomials are in turn called Taylor series.) By using the Taylor series for the exponential function, we can give meaning to the idea of taking e to the power of the differentiation operator. When we do this, it turns out the exponential of the differentiation operator is the shift operator, a bridge between the continuous and discrete worlds.

Dr. Ghrist is a dedicated teacher, and this theorem is a favorite of his from his massive online open course (MOOC), available here. His favorite theorem appears in this video from the MOOC. (You may need to watch previous videos to get an understanding of the notation and nomenclature.) Ghrist’s videos on Taylor series start here, and 3Blue1Brown has a video introduction here.

In each episode of the podcast, we ask our guest to pair their favorite theorem with something: food, beverage, art, music, or other delight in life. Dr. Ghrist chose to pair his theorem with Monster energy drink (“low carb, please, because sugar is not so good for you,” he says). You’ll have to listen to the episode to learn why he thinks it’s the perfect pairing for his favorite theorem.

You can find Dr. Ghrist on his website, YouTube channel, or Twitter. He is working on a series of videos called Calculus Blue that will serve as a video text for multivariable calculus.