After a canter through differential equations and partial differentiation (chapters 1 and 2), I'm now in the chapter 3 foothills of the calculus of variations proper (M820, part of the Open University's maths MSc).
I am very impressed by its subtlety. Richard Feynman said that you haven't understood a topic until you can explain it - briefly! - to an untutored audience. I have been considering how you would explain the CoV to someone without a mathematical background: it's not so easy.
My own understanding is something like this. You start by considering all the different paths to get from some point A to another point B. Each path has some value of a property of interest: it might be the length of the path, it might be the time for an object to slide down from A to B under gravity, and so on. The problem is to find the path which minimises/maximises the value of the property you're interested in. So we might want to find the shortest path, or the least-time curve.
That sets up the problem. The CoV then allows you to explicitly determine the path you're looking for. I haven't figured out how to explain the process explicitly; suffice it to say that the method is a combination of magical, ingenious and deeply non-obvious. To internalise what's going on requires building a whole new paradigm in your head.
And that brings me to quantum mechanics. In parallel I have been revisiting Gary Bowman's excellent "Essential Quantum Mechanics" as preparation for Robert B. Griffiths' "Consistent Quantum Theory" (available in PDF on the web).
Griffiths presents an accessible introduction as to how consistent histories and decoherence can provide an interpretation of quantum mechanics which is less mysterious than the workmanlike Copenhagen interpretation, less nonsensical than consciousness-induced wavefunction collapse, and less extravagant than 'many-worlds'.
I'm quite excited by the prospect!