# MA 227 Standard S12

Divergence Theorem

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

• S12: DivThm. Apply the Divergence Theorem.

## S12: Divergence Theorem

• The Divergence Theorem states that if $$\partial D$$ is the outward-oriented boundary of a solid $$D$$, then $$\iint_{\partial D}\vect{F}\cdot\vect{n}\,d\sigma= \iiint_D\dv\vect F\,dV$$.
• An alternate form of Green’s Theorem holds for boundaries $$\partial R$$ of regions $$R$$ in the plane: $$\int_{\partial R}\vect{F}\cdot\vect{n}\,ds= \iint_R\dv\vect F\,dA$$.
• Note that our theorems all fit the form of a Generalized Fundamental Theorem of Calculus, studied in differential geometry/topology.
• $$\int_{I}f’(x)\,dx=[f]_{\partial I}$$
• $$\int_C\nabla f\cdot d\vect r=[f]_{\partial C}$$
• $$\iint_R\curl\vect{F}\cdot\veck\,dA= \int_{\partial R}\vect{F}\cdot\vect{T}\,ds$$
• $$\iint_R\dv\vect{F}\,dA= \int_{\partial R}\vect{F}\cdot\vect{n}\,ds$$
• $$\iint_S\curl\vect{F}\cdot\vect{n}\,d\sigma= \int_{\partial S}\vect{F}\cdot\vect{T}\,ds$$
• $$\iiint_D\dv\vect{F}\,dV= \iint_{\partial D}\vect{F}\cdot\vect{n}\,d\sigma$$
• General form: $$\underbrace{\int\cdots\iint_{\Omega}}_{n+1}\, f’\,d\omega= \underbrace{\int\cdots\int_{\partial\Omega}}_{n}\, f\,d\omega’$$

### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 15.4 (exercises 5-14, just find flux)
• 15.8 (exercises 5-16)