Question

Hey, I am not sure about the Green's Theorem.
Close loop integral of vector F dot vector dr.
It states that dr is a position vector and equals Unit Tangent multiply by the ds. I am confused about the dr and ds. Are they the same thing? Since unit tangent equals one. So does it mean that dr = ds ?? Thanks..

Answer 1

You're forgetting that dr is a vector while ds is a scalar. ds = |dr|. T, the unit tangent vector to the curve, holds the direction information. So multiplying the unit tangent vector with the magnitude of dr, which is ds, gives you the quantity dr.

Answer 2

ok this is what i was taught
FLUX=F dot nds
CIRCULATION= F dot Tds
now Tds=(dx,dy) "circulation is tedious-Tds)
nds=(dy,-dx)
since these two are perpendicular to each other (simply take the dot product) you can see why flux is = nds and circulation = Tds

flux is the flow to stuff in or out of the area of interest, and circulation is the curl of stuff in the area of interest. Most times when you do examples if the Flux(div) is 0 then there will be curl and same goes for curl=0 then there will be flux