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)

• Greens 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

• Greens Theorem

C
D
6
Stokes Theorem

• A higher dimensional Greens Theorem
• Relates a surface integral over a surface S 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 Counterc
lockwise)
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 Greens 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