1. Stokes’ Theorem Divergence Theorem
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 surface S to a line integral around the boundary curve of S

7. C (boundary has Surface S with boundary C and unit normal vector n n n
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 solid E 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