16.8 and 16.9

1. 16.8 and 16.9 Stokes’ Theorem Divergence Theorem

2. Important Theorems we know • Fundamental theorem of Calculus a b

3. Important Theorems we know • Fundamental theorem of Calculus a b • Fundamental Theorem of Line Integrals r(b) r(a)

4. Important Theorems we know • Fundamental theorem of Calculus a b • Fundamental Theorem of Line Integrals r(b) r(a) • Green’s Theorem C D

5. Important Theorems we know • Fundamental theorem of Calculus a b • Fundamental Theorem of Line Integrals r(b) r(a) Relate an integral of a “derivative” to the original function on the boundary • Green’s Theorem C D

6. Stokes’ Theorem • A higher dimensional Green’s Theorem • Relates a surface integral over a surfaceS to a line integral around the boundary curve of S

7. Surface S with boundary C and unit normal vector n n n C (boundary has a positive orientations: Counterclockwise)

8. Stokes’ Theorem Let S be an oriented piecewise smooth surface that is bounded by a simple, closed piecewise-smooth boundary curve C with positive orientation. Let F be a vector field whose components have continuous first partial derivatives on R3. Then

9. Stokes’ Theorem: A closer look

10. Example

11. The Divergence Theorem • An extension of Green’s Theorem to 3-D solid regions • Relates an integral of a derivative of a function over a solidE to a surface integral over the boundary of the solid.

12. The Divergence Theorem Let E be a simple solid region and let S be the boundary surface of E, given with positive orientation. Let F be a vector field whose components have continuous first partial derivatives on an open region containing E. Then

13. Example