MA 242.003

1. MA 242.003 Day 68: April 22, 2013 Green’s theorem example 13.7: Stokes’ theorem examples 13.8: The Divergence Theorem

2. 13.4: Green’s Theorem Let C be a positively oriented, piecewise-smooth, simple closed curved in the plane, and let D be the region bounded by C. If P and Q have continuous partial derivativeson an open region containing D, then

3. Application of Green’s theorem: Area of a plane region QUESTION: Since the AREA of a plane region D is given by the double integral of f(x,y) = 1 over D, can we choose P(x,y) and/or Q(x,y) in Greens’ theorem to give the area of D?

4. Application of Green’s theorem: Area of a plane region QUESTION: Since the AREA of a plane region D is given by the double integral of f(x,y) = 1 over D, can we choose P(x,y) and/or Q(x,y) in Greens’ theorem to give the area of D? ANSWER:Yes, choose Q = x and P=0, so then

5. Application of Green’s theorem: Area of a plane region QUESTION: Since the AREA of a plane region D is given by the double integral of f(x,y) = 1 over D, can we choose P(x,y) and/or Q(x,y) in Greens’ theorem to give the area of D? ANSWER:Yes, choose Q = x and P=0, so then

6. Example: Compute the area of a circle of radius a Parameterization of the circle:

7. Example: Compute the area of a circle of radius a Area = Parameterization of the circle:

10. (continuation of example)

12. (continuation of example)

15. Section 13.7 The Divergence Theorem of Gauss

16. Section 13.7 The Divergence Theorem of Gauss Relates a flux integral of a vector field to the volume integral of the divergence of that vector field.

17. First we need a definition:

18. First we need a definition: Definition: A solid 3-dimsional region is SIMPLE if it can be described as a type 1, type 2 and a type 3 region in space.

19. First we need a definition: Definition: A solid 3-dimsional region is SIMPLE if it can be described as a type 1, type 2 and a type 3 region In space.

21. Note: E and S are uniquely related to each other, unlike the relationship between S and C in Stokes’ theorem.

22. The divergence theorem is used, for example, in electrostatics, where one encloses a region inside a “Gaussian pillbox” as in the example below:

23. Here is a clip from Wikipedia which discusses “valid” and “invalid” Gaussian surfaces.

24. Here is a clip from Wikipedia which discusses “valid” and “invalid” Gaussian surfaces. A valid surface must enclose a 3-dimensional region E.

25. Remark about problem STATEMENTS:

26. Remark about problem STATEMENTS: 1. If a problem tells you to “USE THE DIVERGENCE THEOREM to compute

27. Remark about problem STATEMENTS: 1. If a problem tells you to “USE THE DIVERGENCE THEOREM to compute then you should compute

28. Remark about problem STATEMENTS: 1. If a problem tells you to “USE THE DIVERGENCE THEOREM to compute then you should compute 2. If a problem tells you to “USE THE DIVERGENCE THEOREM to compute

29. Remark about problem STATEMENTS: 1. If a problem tells you to “USE THE DIVERGENCE THEOREM to compute then you should compute 2. If a problem tells you to “USE THE DIVERGENCE THEOREM to compute then you should compute

31. (continuation of example)

33. (continuation of example)