Green's Theorem, also known as the Stokes' Theorem for the Plane, relates the line integral of a vector field around a simple closed curve to the double integral of the curl of the same vector field over the area enclosed by the curve. It is an important theorem in vector calculus and has applications in many fields, including physics and engineering.
The statement of Green's Theorem is as follows:
Let C be a simple closed curve in the plane, oriented counterclockwise, and let D be the region enclosed by C. Let F = P(x,y)i + Q(x,y)j be a vector field whose components have continuous first partial derivatives in an open region containing D. Then,
∮C F · dr = ∬D (∂Q/∂x - ∂P/∂y) dA
where ∮C denotes the counterclockwise line integral around C, dr is a differential element of arc length along C, ∂Q/∂x and ∂P/∂y are the partial derivatives of Q and P with respect to x and y, respectively, and dA is a differential element of area in the xy-plane.
Proof:
The proof of Green's Theorem can be broken down into three steps:
Step 1: Consider a rectangle R whose sides are parallel to the coordinate axes and which contains D in its interior. Divide R into small rectangles, and let Δx and Δy denote the lengths of the sides of each small rectangle. Let C denote the boundary of R, oriented counterclockwise. Then by the linearity of line integrals,
∮C F · dr = ∑ ∮Ci F · dr,
where Ci denotes the boundary of the ith small rectangle.
Step 2: For each small rectangle, apply the Fundamental Theorem of Calculus for line integrals to obtain
∮Ci F · dr = ∫bi ai (∂Q/∂x - ∂P/∂y) dt,
where ai and bi are the endpoints of the ith side of the rectangle, oriented in the counterclockwise direction.
Step 3: Summing over all the small rectangles and taking the limit as the size of the rectangles approaches zero yields
∮C F · dr = ∬D (∂Q/∂x - ∂P/∂y) dA,
which is the desired result.
Therefore, Green's Theorem is proved.
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