[[Noteworthy Uses]]:: *[[Noteworthy Uses]]* [[Proved By]]:: *[[Proved By|Crucial Dependencies]]* Intuition:: *[[Intuition]]* Specializations:: *[[Specializations]]* Generalizations:: [[Generalized Stokes' Theorem]] ---- - Let $M$ be a $2$-[[differentiable Euclidean submanifold (with or without boundary)|manifold]] in $\rr ^{2}$; - Let $\omega = P \ dx + Q \ dy$ be an arbitrary $1$-form defined on an [[open set]] containing $M$. - To relate to [[vector field]]s, recall [[div grad diagram]] and [[div grad curl diagram]] (musical isomorphisms). > [!theorem] Theorem. ([[Green's Theorem]]) > Then $\int _{M} \left( \frac{ \partial Q }{ \partial x } - \frac{ \partial P }{ \partial y } \right) = \int _{\partial{M}} P \ dx + Q \ dy .$ > [!proof]- Proof. ([[Green's Theorem]]) > We compute $d\omega = (\frac{ \partial Q }{ \partial x } - \frac{ \partial P }{ \partial y }) \ dx \wedge dy$ . Now by [[Generalized Stokes' Theorem]] the result follows. ---- #### ----- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as Tag > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```