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MA 242.003

MA 242.003 . Day 57 – April 8, 2013 Section 13.5: Review Curl of a vector field Divergence of a vector field. Section 13.5 Curl of a vector field. “A way to REMEMBER this formula”. “A way to REMEMBER this formula”. “A way to REMEMBER this formula”. “A way to REMEMBER this formula”.

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MA 242.003

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  1. MA 242.003 • Day 57 – April 8, 2013 • Section 13.5: • Review Curl of a vector field • Divergence of a vector field

  2. Section 13.5Curl of a vector field

  3. “A way to REMEMBER this formula”

  4. “A way to REMEMBER this formula”

  5. “A way to REMEMBER this formula”

  6. “A way to REMEMBER this formula”

  7. “A way to REMEMBER this formula”

  8. “A way to REMEMBER this formula”

  9. (continuation of example)

  10. Let F represent the velocity vector field of a fluid. What we find is the following: Example: F = <x,y,z> is diverging but not rotating curl F = 0

  11. 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

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

  13. 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>

  14. (See Maple worksheet for the calculation)

  15. Differential Identity involving curl

  16. 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”

  17. The Divergence of a vector field

  18. The Divergence of a vector field

  19. The Divergence of a vector field

  20. The Divergence of a vector field

  21. The Divergence of a vector field Then div F can be written symbolically as:

  22. The Divergence of a vector field Then div F can be written symbolically as:

  23. The Divergence of a vector field

  24. The Divergence of a vector field

  25. So the vector field

  26. So the vector field Is incompressible

  27. So the vector field Is incompressible However the vector field

  28. So the vector field Is incompressible However the vector field Is NOT – it is diverging!

  29. Differential Identity involving div

  30. Differential Identity involving div

  31. Differential Identity involving div Proof:

  32. (continuation of proof)

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