Final answer:
To evaluate a line integral using Green's Theorem, one must compute the partial derivatives of the vector field components and perform a double integral over the region enclosed by the curve.
Step-by-step explanation:
The student's question is about applying Green's Theorem to a vector field f{F}(x, y) to evaluate a line integral. Unfortunately, the details of the vector field were not fully provided, making it challenging to give a definitive answer. However, I can explain the concept generally. Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D enclosed by C. The theorem is expressed as:
∬C (P dx + Q dy) = ∫∫D (∂Q/∂x - ∂P/∂y) dA
where P and Q are the components of the vector field F(x, y).
To use Green's Theorem, one must ensure that the curve C is positively oriented, the field is continuously differentiable, and region D is simply connected. The process involves calculating the partial derivatives of P and Q with respect to x and y, respectively, and then evaluating the double integral over the region D for ∂Q/∂x - ∂P/∂y.