George Green was a brilliant mathematician who also operated a windmill in Nottingham. I used to cycle past it almost every day. I never understood what his big trick was. I think I've got it at last. It is to do with vector fields. An example of a vector field is a wind speed weather map. It is a series of straight arrows drawn on the map showing the direction of the wind and the length of the arrow representing wind strength. Entry-level differentiation is a mathematical trick to find the gradient of a graph. But how do you differentiate a set of arrows? There are two techniques which might have something to do with vectors having size and direction. These techniques are call div (divergence) and curl. Curl turns out to be the circulation per unit area of the fluid or whatever else is modeled by the vector field. I've drawn a lot of mini circulations inside the Helmholtz coil shown above. To find the actual circulation, we need to do the curl of the function (let's call it F for field) and multiply it be the tiny area because curl is circulation per unit area. So each square above represents a circulation curl F x dA. If we add them all up we get the total circulation. But look at the picture below. At each internal boundary the flow is equal and opposite and thus cancels. Adding up all the circulation elements just gives you the large total flow, shown by the outer red arrow in the top picture. Green showed that this outer flow could be found by adding up tiny sections of the line denoted dl multiplied by the function F. Thus adding up F.dl all the way round a curved line equals adding up all the curlF x dA across an area. He links function F round a circular line with the curl of F across an area inside this curve. This can be useful trick to swap between lines and areas
