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Exploring Line Integrals and Curl of Vector Fields in Calculus

This content focuses on the fundamental theorem for line integrals and the concept of curl in vector fields, illustrated with examples. Key topics include the behavior of velocity vector fields, distinguishing between divergent and rotational fields, and understanding simply-connected regions. Important formulas are presented, alongside mnemonic aids for remembering them. The discussion includes graphical representations from Maple, reinforcing the concepts of irrotational fields and differential identities involving curl. This entry combines theory with practical examples for enhanced comprehension.

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Exploring Line Integrals and Curl of Vector Fields in Calculus

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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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