Final answer:
The question asks for the application of Stokes' theorem to calculate the line integral of a vector field along a certain curve without actually evaluating the integral. The process involves finding the curl of the vector field and setting up the surface integral over a surface bounded by the curve.
Step-by-step explanation:
The question involves the application of Stokes' theorem to calculate the line integral of a vector field F(x,y,z) along a curve C, which is the intersection of a plane and a cylinder in R³. According to Stokes' theorem, the line integral of a vector field around a closed curve can be transformed into a surface integral of the curl of the vector field over a surface bounded by the curve. In this scenario, we are oriented counter-clockwise when viewed from above the surface.
To apply Stokes' theorem, we first need to calculate the curl of F, which involves taking the partial derivatives of the components of F. Then, we would define a surface S bounded by the curve C, and apply the curl of F to this surface integral. The actual evaluation of the integral is not requested, as the goal is only to set up the expression using Stokes' theorem.
Since the evaluation of the integral isn't required, we do not perform the calculations but rather ensure that the student understands the process to apply Stokes' theorem appropriately to convert the line integral into a surface integral of the curl of the vector field over the surface S.