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)