# MA 227 Standard S11

Green's Theorem and Stokes's Theorem

At the end of the course, each student should be able to…

• S11: GreenStokes. Apply Green’s Theorem and Stokes’s Theorem.

## S11: Green’s Theorem and Stokes’s Theorem

• Stokes’s Theorem states that if the boundary $$\partial S$$ of a surface $$S$$ is oriented counter-clockwise with respect to the orientation of $$S$$, then $$\int_{\partial S}\vect F\cdot d\vect{r}= \iint_S\curl\vect F\cdot\vect{n}\,d\sigma$$.
• This simplifies to Green’s Theorem when $$S$$ is the region $$R$$ in the plane oriented by $$\veck$$ with counter-clockwise boundary $$C$$: $$\int_C \vect F\cdot d\vect{r}=\iint_R \curl\vect F\cdot\veck\,dA = \iint_R(N_x-M_y)\,dA$$.

### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 15.4 (exercises 5-14, just find circulation)
• 15.7 (exercises 1-6)