91Ó°ÊÓ

Green's Theorem and Stokes' Theorem .

$$\begin{gathered} \mathrm{In}\mathrm{stokes}\text{'}\mathrm{s}\mathrm{theoram},\mathrm{if}\mathrm{S}\mathrm{is}\mathrm{the}\mathrm{region}\mathrm{of}\mathrm{the}\mathrm{xy}-\mathrm{plane},\mathrm{then}, \\ \mathrm{n}=\mathrm{k},\mathrm{and}, \\ \mathrm{curl}\mathrm{F}=\left(\frac{∂{\mathrm{F}}_2}{∂\mathrm{x}}-\frac{∂{\mathrm{F}}_1}{∂\mathrm{y}}\right)\mathrm{k}. \\ \mathrm{Then},\mathrm{by}\mathrm{Stokes}\text{'}\mathrm{theoram}, \\ {∫}_{\mathrm{c}}\mathrm{F}.\mathrm{dr}=∫{∫}_{\mathrm{s}}\mathrm{curl}\mathrm{F}.\mathrm{ndS} \\ =∫{∫}_{\mathrm{s}}\left(\frac{∂{\mathrm{F}}_2}{∂\mathrm{x}}-\frac{∂{\mathrm{F}}_1}{∂\mathrm{y}}\right)\mathrm{k}.\mathrm{kdS} \\ =∫{∫}_{\mathrm{s}}\left(\frac{∂{\mathrm{F}}_2}{∂\mathrm{x}}-\frac{∂{\mathrm{F}}_1}{∂\mathrm{y}}\right)\mathrm{dS} \\ =∫{∫}_{\mathrm{R}}\left(\frac{∂{\mathrm{F}}_2}{∂\mathrm{x}}-\frac{∂{\mathrm{F}}_1}{∂\mathrm{y}}\right)\mathrm{dA} \\ \mathrm{This}\mathrm{implies}\mathrm{that},\mathrm{Green}\text{'}\mathrm{s}\mathrm{theoram}\mathrm{is}\mathrm{a}\mathrm{special}\mathrm{case}\mathrm{of}\mathrm{stokes}\text{'}\mathrm{s}\mathrm{theoram}. \end{gathered}$$

Green's Theorem relates only to two-dimensional vector fields and to regions in the two dimensional plane. Stokes' Theorem generalizes Green's Theorem to three dimensions. Hence, generalize Green's Theorem to regions that are two-dimensional but that do not lie in the xy-plane. Then, Stokes' Theorem can be interpreted as a generalization of Green's Theorem traditionally written in terms of the curl of a vector field.

Hence, Green's Theorem is a special case of Stokes Theorem.