Find $\int_c ydx+zdy+xdz~$ where $C$ is the curve of the intersection of the two surfaces $z=xy$ and $x^2+y^2=1$ , traversed once in a direction that appears counterclockwise when viewed from high above the $xy$ plane. ( Without using Stokes Theorem)
Solution Attempt:
I seem to have no idea how to represent the curve of the intersection of the two surfaces $z=xy$ and $x^2+y^2=1$ in mathematical terms.
However, here's what I tried with the given equations : Differentiating them :
$dz = x dy+y dx$ and $xdx+ydy=0$ .
I thought of substituting the above results back into the integrand, but this doesn't seem a very good idea.
Could anyone tell me how do I proceed from here?
Thank you for reading through!
Best Answer
I think I got it. Just struck me that $x=\cos \theta, y =\sin \theta, z = \sin (2 \theta)/2$