Final answer:
To evaluate the line integral ∮ F · dr along a curve C, one must know details about the field and curve. Depending on the context, this can be done using Green's theorem, Stokes' theorem, or the Divergence theorem, all of which are theorems in vector calculus used under specific conditions.
Step-by-step explanation:
The question asks to evaluate the line integral ∮ F · dr along a curve C using one of the theorems. To answer this correctly, we need to know the field F and the curve C, as well as any other given information about the region enclosed by the curve. Among the choices given, Green's theorem, Stokes' theorem, and the Divergence theorem all relate to the evaluation of line integrals, but they are used in specific contexts:
- Green's theorem is applicable for planar regions and relates a line integral around a simple closed curve C to a double integral over the region D bounded by C.
- Stokes' theorem generalizes Green's theorem to three dimensions, relating a line integral around a closed curve C in space to a surface integral over a surface S bounded by C.
- The Divergence theorem relates the flux of a vector field through a closed surface to the volume integral of the divergence of the field over the volume enclosed by the surface.
Without the specific details of the vector field and the curve, we cannot definitively use any of the given theorems to evaluate the integral. If the problem relates to evaluating a line integral around a closed planar curve C, then Green's theorem would be the appropriate choice. If the curve C is space and bounds a surface, Stokes' theorem could be relevant. If we were concerned with the flux through a closed surface, the Divergence theorem would be the correct pick.