Statement of Green's Theorem

Calculus 3 · Axiom Academy

A bridge connecting a line integral around a closed curve to a double integral over the region it encloses. Let C be a positively oriented , piecewise-smooth, simple closed curve enclosing a region R . Then the line integral around C equals the double integral of the curl over R : Here C is traversed counterclockwise (positive orientation, region on your left), P(x,y) and Q(x,y) are the components of the field , and R is the region enclosed by C . The integrand measures the microscopic circulation — the scalar curl — of the vector field at each point. 3. Proof Sketch: Rectangular Grid The key idea is to divide the region R into small rectangles, apply the theorem to each one, and add up the results. The magic: every interior edge is shared by two rectangles and is traversed twice in opposite directions , so those contributions cancel. Only the outer boundary survives. Green's Theorem is a powerful computational tool that often turns a hard calculation into an easy one: Area formula: setting P = -y and Q = x gives . Path independence: if everywhere, then for every closed curve. Flux & flow: computing flow across a boundary from interior properties. Foundation for Stokes' Theorem: the same idea generalizes to curved surfaces in 3D. You've seen how a walk around a closed boundary and a sweep across the region it encloses are two ways of computing the very same number. Scroll up to revisit any step.

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