Final answer:
Green's theorem allows the conversion of a line integral around a closed curve into a double integral over the region bounded by the curve, often simplifying the evaluation.
Step-by-step explanation:
The student's question is asking how to evaluate a line integral using Green's theorem. Green's theorem relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C. This theorem is a special case of the more general Stokes' theorem. When evaluating the line integral ∫ C f⋅dr, one would typically convert this into a double integral over the domain D enclosed by the curve C, applying Green's theorem. An important aspect of applying this theorem is to ensure the vector field is defined and differentiable on an open region containing D and its boundary curve C.
To use Green's theorem to evaluate a line integral, we first need to make sure that our function f can be written as M dx + N dy, where M and N have continuous partial derivatives over the region D. The theorem then states that ∫ C (M dx + N dy) = ∫∫ D (∂N/∂x - ∂M/∂y) dA, where the right-hand side represents the double integral over the domain D.
In summary, using Green's theorem to evaluate a line integral converts the problem into computing a double integral over the region enclosed by the curve, which is often more convenient.
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