[[Noteworthy Uses]]:: *[[Noteworthy Uses]]* [[Proved By]]:: *[[Proved By|Crucial Dependencies]]* ---- - Let $M$ be a $2$-[[differentiable Euclidean submanifold (with or without boundary)|manifold]] in $\rr ^{3}$; - Let $\omega = P \ dx + Q \ dy + R \ dz$ 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. ([[Classical Stokes' Theorem]]) > Then $\int _{M} (\text{curl }F \cdot N) \, \d V = \int _{\partial{M}} (F \cdot T) \ \d V. $ > [!proof]- Proof. ([[Classical Stokes' Theorem]]) > Compute $d\omega=\left( \frac{ \partial R }{ \partial y } - \frac{ \partial Q }{ \partial z } \right) \ dy \wedge dz - \left( \frac{ \partial P }{ \partial z } - \frac{ \partial R }{ \partial x } \right) \ dx \wedge dz + \left( \frac{ \partial Q }{ \partial x } - \frac{ \partial Q }{ \partial y } \right) \ dx \wedge dy$. Notice that if we set $F:=(P,Q,R)$ then these coefficients are from the [[curl]] definition, i.e., we really have $d\omega = (\text{curl }F)_{1} \ dy \wedge dz - (\text{curl }F)_{2} \ dx \wedge dz + (\text{curl }F)_{3} \ dx \wedge dy.$ By [[Generalized Stokes' Theorem]] we get that $\int _M d\omega = \int _{\partial{M}} (F \cdot T) \ \d V$, and then in replacing the LHS via [[circulation integral]] 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 > ```