vela said:You miscalculated dQ/dx.
Green's theorem is a fundamental theorem in the field of vector calculus that relates the line integral of a vector field around a closed curve to the double integral of the curl of the vector field over the region enclosed by the curve.
Green's theorem can be applied to the boundary of a rectangle by breaking the region into smaller rectangles and using the theorem to calculate the line integrals along each boundary. Then, the line integrals along the common boundaries will cancel out, leaving only the line integrals along the outer boundary of the original rectangle.
The formula for Green's theorem on a rectangle is: ∫C Pdx + Qdy = ∬R (∂Q/∂x - ∂P/∂y)dA, where C is the boundary of the rectangle, P and Q are the components of the vector field, and R is the region enclosed by the rectangle.
Green's theorem can be applied to any shape as long as the region can be divided into smaller, simpler shapes for which the theorem can be applied. This is because the theorem relies on the cancellation of line integrals along common boundaries.
Some practical applications of Green's theorem include calculating work done by a conservative force, calculating flux through a region, and evaluating line integrals in physics and engineering problems. It is also used in fluid dynamics to calculate flow rates and circulation around a closed path.