Green’s Theorem

Green’s Theorem establishes a relationship between a line integral around a closed plane curve and a double integral over the region enclosed by that curve. It applies to a vector field F(x, y) = 〈P(x, y), Q(x, y)〉, where P and Q have continuous first partial derivatives on an open set containing the region.

Let C be a positively oriented, simple, closed, piecewise smooth curve, and let R be the plane region bounded by C. Green’s Theorem states that

\begin{equation*}\oint_C P\,dx+Q\,dy =\iint_R \liparens{\jfrac{\partial Q}{\partial x}-\jfrac{\partial P}{\partial y}}dA\end{equation*}

The left side is a line integral taken around the boundary C, while the right side is a double integral over the region R.

The expression

\begin{equation*}\jfrac{\partial Q}{\partial x}-\jfrac{\partial P}{\partial y}\end{equation*}

is the two-dimensional scalar curl of the vector field. It measures the infinitesimal rotational behavior of the field at each point in R. Green’s Theorem states that the total boundary circulation equals the integral of the curl over the entire region.

Positive orientation means that C is traversed counterclockwise, so the region R remains on the left side of the direction of motion. The theorem requires C to be closed and piecewise smooth, and R must be a suitable plane region. The functions P and Q must be continuously differentiable on a domain containing R. Under these assumptions, Green’s Theorem converts a boundary integral into an equivalent area integral.