Use Stokes’ theorem to solve the following integral (the curve is oriented counterclockwise when viewed from above.) $$\int_C(x+ 2y)dx+ (2z+ 2x)dy+ (z+y)dz$$ where $C$ is the intersection of the sphere $x^2 + y^2 +z^2= 1$ and the plane $y = z$
Can someone guide me on how to answer this? Thank you.
According to Stokes' theorem and with $\vec F(x,y,z) = \left( x+2y,2z+2x,z+y \right)$, you have: $$\oint_C \vec F(x,y,z) \cdot \mbox{d}\vec r = \iint_S \nabla \times \vec F \cdot \mbox{d}\vec S$$ The curl of $\vec F$ is very easy: $\nabla \times \vec F = (-1,0,0)$, can you continue with the surface integral?
Hint: for $S$, you can choose any surface with the given border $C$ as boundary. Choose wisely.