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MA 242.003

MA 242.003 . Day 55 – April 5, 2013 Section 13.3: The fundamental theorem for line integrals An interesting example Section 13.5: Curl of a vector field. x = cos(t ) y = sin(t ) t = 0 .. Pi. x = cos(t ) y = - sin(t ) t = 0 .. Pi. D open.

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MA 242.003

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  1. MA 242.003 • Day 55 – April 5, 2013 • Section 13.3: The fundamental theorem for line integrals • An interesting example • Section 13.5: Curl of a vector field

  2. x = cos(t) y = sin(t) t = 0 .. Pi

  3. x = cos(t) y = - sin(t) t = 0 .. Pi

  4. D open Means does not contain its boundary:

  5. D open Means does not contain its boundary:

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

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

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

  9. Section 13.5Curl of a vector field

  10. Section 13.5Curl of a vector field

  11. Section 13.5Curl of a vector field

  12. Section 13.5Curl of a vector field

  13. Section 13.5Curl of a vector field

  14. “A way to REMEMBER this formula”

  15. “A way to REMEMBER this formula”

  16. “A way to REMEMBER this formula”

  17. “A way to REMEMBER this formula”

  18. “A way to REMEMBER this formula”

  19. “A way to REMEMBER this formula”

  20. (see maple for sketch)

  21. (see maple for sketch)

  22. All of these velocity vector fields are ROTATING.

  23. All of these velocity vector fields are ROTATING. What we find is the following:

  24. All of these velocity vector fields are ROTATING. What we find is the following: Example: F = <x,y,z> is diverging but not rotating curl F = 0

  25. All of these velocity vector fields are ROTATING. What we find is the following: F is irrotational at P. Example: F = <x,y,z> is diverging but not rotating curl F = 0

  26. All of these velocity vector fields are ROTATING. What we find is the following:

  27. All of these velocity vector fields are ROTATING. What we find is the following: Example: F = <-y,x,0> has non-zero curl everywhere! curl F = <0,0,2>

  28. Differential Identity involving curl

  29. Differential Identity involving curl Recall from the section on partial derivatives: We will need this result in computing the “curl of the gradient of f”

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