What is the simple relationship expressed verbally between flux, circulation, div, and curl, as captured by Green's, Stokes', and Gauss' Theorems? Below is what I've been able to assemble: Can you confirm or improve it?
Given a vector field, we can measure its instantaneous rate of change in multiple ways, the two most important being div (a scalar) and curl (a vector). Each of these can be accumulated in a region. These theorems shows that this accumulation equals a property of the region's boundary:
Flux (through a boundary) equals the accumulation of divergence (in the bounded region).
Circulation (around a boundary) equals the accumulation of curl (in the bounded region).
More fundamentally, neither flux nor circulation is limited to boundaries: they, or their equivalents, can be measured in non-closed curves as well. Only in that case, there is no relationship to any accumulation. That is: Flux can be measured across a simple curve or surface that is not closed. But this has no relationship to divergence.
Likewise, a quantity equivalent to circulation can be measured through a simple curve or surface that is not closed, but if the curve isn't closed, it's called a line integral or work integral (or, for a surface, a surface integral). Neither has any relationship to curl.