#MTBoS: There is a beautiful way to teach the fundamental theorems of vector #calculus! In this thread, I'll outline a natural, intuitive approach. References at the end. Thanks to @Quasilocal for starting this conversation.

1. In single variable calculus, we define the derivatives as an instantaneous rate of change, via a limit. We can define curl and div analogously as measures of rotation (circulation density) and expansion (flux density).

2. The FTC tells us "sum of local change = global change." Green's thm is an analogue with two versions: "sum of local circ. = global circ." and "sum of local flux = global flux," which generalize to Stokes' and Gauss's theorems, respectively.