Determine whether the following statements are true and give an explanation or counterexample.

a. The work required to move an object around a closed curve C in the presence of a vector force field is the circulation of the vector field on the curve.

b. If a vector field has zero divergence throughout a region (on which the conditions of Green’s Theorem are met), then the circulation on the boundary of that region is zero.

c. If the two-dimensional curl of a vector field is positive throughout a region (on which the conditions of Green’s Theorem are met), then the circulation on the boundary of that region is positive (assuming counterclockwise orientation).