Find value of a line integral using Stokes' theorem

Question

Given a vector field $\vec A = y\vec i + z\vec j + x\vec k$ Use Stokes' formula to calculate the line integral $\oint\limits_C {\vec Ad\vec r} $ in here, $C$ is a circle ${x^2} + {y^2} + {z^2} = {a^2}$ intersects $x + y + z = 0$

Answer 1

The curl of your vector field is $-{\bf i}-{\bf j}-{\bf k}$ which is normal to the plane $x+y+z=0$, the length of the vector is $\sqrt{3}$, so the integral is $\pm \sqrt{3}$ the area of the disk that the circle bounds in the plane. Unfortunaley you didn't give the orientation of the circle to have a well defined answer, so I can't tell you which one.