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MA 242.003

MA 242.003 . Day 55 – April 4, 2013 Section 13.3: The fundamental theorem for line integrals Review theorems Finding Potential functions The Law of Conservation of Total Energy. Section 13.3 The Fundamental Theorem for Line Integrals. In which we characterize conservative vector fields.

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MA 242.003

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  1. MA 242.003 • Day 55 – April 4, 2013 • Section 13.3: The fundamental theorem for line integrals • Review theorems • Finding Potential functions • The Law of Conservation of Total Energy

  2. Section 13.3 The Fundamental Theorem for Line Integrals In which we characterizeconservative vector fields And generalize the FTC formula

  3. Note that we now have one characterization of conservative vector fields on 3-space. They are the only vector fields whose line integrals are independent of path.

  4. Note that we now have one characterization of conservative vector fields on 3-space. They are the only vector fields whose line integrals are independent of path. Unfortunately, this characterization is not very practical!

  5. We proved: This is another characterization of conservative vector fields!

  6. We proved: This is another characterization of conservative vector fields! The question arises: Is the CONVERSE true?

  7. We proved: This is another characterization of conservative vector fields! YES! The question arises: Is the CONVERSE true?

  8. Proof given after we study Stokes’ theorem in section 13.7.

  9. FINDING POTENTIAL FUNCTIONS QUESTION: Now that we have a conservative vector field,

  10. FINDING POTENTIAL FUNCTIONS QUESTION: Now that we have a conservative vector field, how do we find potential functions?

  11. FINDING POTENTIAL FUNCTIONS QUESTION: Now that we have a conservative vector field, how do we find potential functions? SOLUTION: Integrate the three equations, one at a time, to find the potentials for F.

  12. Illustration of the method: F = < 2x + z , 2y + z, 2z + x + y> Conservative?

  13. Illustration of the method: F = < 2x + z , 2y + z, 2z + x + y> Find potential functions:

  14. (continuation of example)

  15. You can construct your own “find the potential functions” as follows:

  16. You can construct your own “find the potential functions” as follows: 1. Choose a function f(x,y,z) . For example:

  17. You can construct your own “find the potential functions” as follows: 1. Choose a function f(x,y,z) . For example:

  18. You can construct your own “find the potential functions” as follows: 1. Choose a function f(x,y,z) . For example: 2. Then compute its gradient:

  19. You can construct your own “find the potential functions” as follows: 1. Choose a function f(x,y,z) . For example: 2. Then compute its gradient: Now you have a conservative vector field – so find its potential functions (you already know the answer!).

  20. An Application: The Law of Conservation of Total Energy

  21. An Application: The Law of Conservation of Total Energy t=a t=b

  22. We calculate the work done in two different ways. t=a t=b

  23. We calculate the work done in two different ways. t=a t=b

  24. We calculate the work done in two different ways. t=a t=b

  25. An Identity: We can derive a very useful identity by differentiating the function

  26. We calculate the work done in two different ways. t=a t=b

  27. t=a We calculate the work done in two different ways. t=b

  28. t=a We calculate the work done in two different ways. t=b

  29. t=a We calculate the work done in two different ways. t=b

  30. Red curve is Kinetic energy K Blue curve is gravitational potential energy U Green curve is the Total Energy E = K + U

  31. D open Means does not contain its boundary:

  32. D open Means does not contain its boundary:

  33. D simply-connected means that each closed curve in D contains only points in D.

  34. D simply-connected means that each closed curve in D contains only points in D. Simply connected regions “contain no holes”.

  35. D simply-connected means that each closed curve in D contains only points in D. Simply connected regions “contain no holes”.

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