What does the result of Green's Theorem represent?

Question

I know how to use the Green's Theorem, but I do not know what the answer mean.

Answer 1

green's theorem shows that instead of using a line integral... you could use a double integral (for area) as a simpler way of getting to the value...

§

Answer 2

If you're looking for an intuitive conception, you can think of the line integral around a region (which is sometimes called the circulation) as being a property of the region itself rather than its boundary. This makes sense, because if you take two adjacent regions and find the circulation around their union, it will just be the sum of the circulation around each region.

Taking this view, it would make sense that the circulation around one large region could be found by dividing it into infinitesimal components and integrating the circulation around each component. Green's theorem is then nothing more than the assertion that the circulation around an infinitesimal component divided by its area happens to be the curl of the function at that point. Or stated slightly more formally (but just as imprecisely), it states that the differential of circulation is the curl.

It should be noted that the idea I just used, of dividing the large region into little regions and then adding up the circulation around all of them is actually used in the formal proof of Green's theorem. Basically, Green's theorem is proven for a few types of "well-behaved" regions (specifically, regions for which the limits of integration can be written as a

Answer 3

Just like any integral, it represents area. This integral represents the enclosed area of the curve. However, because of its relation to curl, it represents the signed area.