Vector Fields Acting on a Curve - PowerPoint PPT Presentation

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Vector Fields Acting on a Curve

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  1. VC.04-VC.08 Redux Vector Fields Acting on a Curve

  2. VC.04: Find a surface with the given gradient (Ex 1 From VC.04 Notes)

  3. VC.04: Gradient Fields

  4. VC.04: Gradient Fields

  5. VC.04: Breaking Field Vectorsinto Two Components

  6. VC.04: Flow Along and Flow Across

  7. VC.05: Measuring the Flow of a Vector Field ALONG a Curve These integrals are usually called line integrals or path integrals.

  8. VC.05: Measuring the Flow of a Vector Field ALONG a Curve

  9. VC.05: Measuring the Flow of a Vector Field Along Another Closed Curve

  10. VC.05: Measuring the Flow of a Vector Field Along Another Closed Curve  Counterclockwise

  11. VC.05: Measuring the Flow of a Vector Field ACROSS a Curve

  12. VC.05: Measuring the Flow of a Vector Field Across Another Closed Curve  Counterclockwise

  13. VC.05: The Gradient Test

  14. VC.05: The Flow of a Gradient Field Along a Closed Curve

  15. VC.05: Path Independence: The Flow of a Gradient Field Along an Open Curve

  16. VC.06: The Gauss-Green Formula

  17. VC.06: Measuring the Flow of a Vector Field ALONG a Closed Curve Let C be a closed curve with a counterclockwise parameterization. Then the net flow of the vector field ALONG the closed curve is measured by: Let region R be the interior of C. If the vector field has no singularities in R, then we can use Gauss-Green:

  18. VC.06: Summary: The Flow of A Vector Field ALONG a Closed Curve: • Let C be a closed curve parameterized counterclockwise. Let Field(x,y) be a vector field with no singularities on the interior region R of C. Then: • This measures the net flow of the vector field ALONG the closed curve.

  19. VC.06: Measuring the Flow of a Vector Field ACROSS a Closed Curve Let C be a closed curve with a counterclockwise parameterization. Then the net flow of the vector field ACROSS the closed curve is measured by: Let region R be the interior of C. If the vector field has no singularities in R, then we can use Gauss-Green:

  20. VC.06: Summary: The Flow of A Vector Field ACROSS a Closed Curve: • Let C be a closed curve parameterized counterclockwise. Let Field(x,y) be a vector field with no singularities on the interior region R of C. Then: • This measures the net flow of the vector field ACROSS the closed curve.

  21. VC.06: The Divergence Locates Sources and Sinks Let C be a closed curve with a counterclockwise parameterization with no singularities on the interior of the curve. Then:

  22. VC.06: The Rotation Helps You Find Clockwise/Counterclockwise Swirl Let C be a closed curve with a counterclockwise parameterization with no singularities on the interior of the curve. Then:

  23. VC.06: Avoiding Computation Altogether

  24. VC.06: Find the Net Flow of a Vector Field ACROSS Closed Curve

  25. VC.06: A Flow Along Measurement With a Singularity

  26. VC.06: A Flow Along Measurement With a Singularity

  27. VC.06: A Flow Along With Multiple Singularities? No Problem!

  28. VC.06: Flow Along When rotField(x,y)=0

  29. VC.06: Flow Along When divField(x,y)=0

  30. VC.07: The Area Conversion Factor:

  31. VC.07: The Area Conversion Factor:

  32. VC.07: : Fixing Example 2

  33. VC.07: Mathematica-Aided Change of Variables (Parallelogram Region)

  34. VC.07: Mathematica-Aided Change of Variables (Parallelogram Region)

  35. VC.07: Mathematica-Aided Change of Variables (Parallelogram Region)

  36. VC.08: 3D Integrals:

  37. VC.08: A 3D-Change of Variables with an Integrand