Thread: Contour integration

Hello.

So basically I need to calculate this


where L is defined as


I decided to go for (Wikipedia link below, I assume notation is reasonably understandable)

but, no matter how shameful that is, I am not sure how to quickly define a projection (of the curve) to Oxy plane...

P.S. Moreover, since all this means some specific circulation, how differently should L be described if I wanted it to go the other way (and get the opposite value)?

The easiest way is using the Stokes theorem if you know that.
let , then

Let S be the surface area enclosed by L, then

Let p be the projection to the xy plain then p is a diffeomorphism on S and let q be its inverse, and Let A=p(S), Note that p(x,y,z)=(x,y), so the pullback q*(dx)=dx, q*(dy)=dy

= = area of A = (area of S) *
Where is the angle of the two planes with normal vectors v=(1,1,1) and w=(0,0,1), so
So the anwser is