Jordan Curve | Sahithyan's Notes

A Jordan curve in the plane is a simple closed curve, a closed curve that does not intersect itself except that the initial and terminal points coincide.

Green’s Theorem

Let $C$ be a positively oriented, piecewise smooth Jordan curve in the plane, and let $D$ be the region bounded by $C$ . Suppose

\boldsymbol{F}(x,y) = M(x,y)\hat{i} + N(x,y)\hat{j}

is continuously differentiable on an open region containing $D$ . Then

\oint_C M\,\text{d}x + N\,\text{d}y = \iint_D \left(\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}\right)\text{d}A

Area Formula

Taking $M = -y/2$ and $N = x/2$ gives

A(D) = \iint_D 1\,\text{d}A = \frac{1}{2}\oint_C x\,\text{d}y - y\,\text{d}x

Green’s Theorem for Doubly Connected Regions

Let $R$ be a doubly connected region in the plane, with outer boundary $C_1$ oriented counterclockwise and inner boundary $C_2$ oriented clockwise. If $C_1$ , $C_2$ , and $\boldsymbol{F}(x,y) = M(x,y)\hat{i} + N(x,y)\hat{j}$ satisfy the hypotheses of Green’s theorem, then

\iint_R \left(\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}\right)\text{d}A = \oint_{C_1} M\,\text{d}x + N\,\text{d}y + \oint_{C_2} M\,\text{d}x + N\,\text{d}y