Optimization over points wasn’t enough. The fastest path needed a new idea. Enter the calculus of variations. As discussed in a prior post, the fastest path for a frictionless bead to slide between two points is not a straight line. To find the quickest descent curve, mathematicians had to craft methods to optimize over functions themselves. The gave rise to what is now known as the calculus of variations. In these slides, I unpack the solution. And, more importantly, this lecture introduces concepts in this area of optimization.