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MA 242.003

MA 242.003 . Day 48 – March 21, 2013 Section 12.8: Finish: Triple Integrals in Spherical Coordinates Section 13.1: Vector Fields. Section 12.8 Triple Integrals in Cylindrical Coordinates. Goal : Use spherical coordinates to compute a triple integral that has spherical symmetry. Spheres.

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MA 242.003

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  1. MA 242.003 • Day 48 – March 21, 2013 • Section 12.8: Finish: Triple Integrals in Spherical Coordinates • Section 13.1: Vector Fields

  2. Section 12.8 Triple Integrals in Cylindrical Coordinates Goal: Use spherical coordinates to compute a triple integral that has spherical symmetry. Spheres Cones

  3. Fubini’s theorem for a spherical box:

  4. Generalization to non-spherical boxes

  5. Generalization to non-spherical boxes

  6. Generalization to non-spherical boxes

  7. (Continuation of example)

  8. Chapter 13: Vector Calculus

  9. Chapter 13: Vector Calculus “In this chapter we study the calculus of vector fields,

  10. Chapter 13: Vector Calculus “In this chapter we study the calculus of vector fields, and line integrals of vector fields (work),

  11. Chapter 13: Vector Calculus “In this chapter we study the calculus of vector fields, …and line integrals of vector fields (work), …and the theorems of Stokes and Gauss,

  12. Chapter 13: Vector Calculus “In this chapter we study the calculus of vector fields, …and line integrals of vector fields (work), …and the theorems of Stokes and Gauss, …and more”

  13. Section 13.1: Vector Fields

  14. Section 13.1: Vector Fields

  15. Section 13.1: Vector Fields Wind velocity vector field 2/20/2007

  16. Section 13.1: Vector Fields Wind velocity vector field 2/21/2007 Wind velocity vector field 2/20/2007

  17. Section 13.1: Vector Fields Ocean currents off Nova Scotia

  18. Section 13.1: Vector Fields Airflow over an inclined airfoil.

  19. General form of a 2-dimensional vector field

  20. General form of a 2-dimensional vector field

  21. General form of a 2-dimensional vector field

  22. General form of a 2-dimensional vector field

  23. General form of a 2-dimensional vector field Examples:

  24. General form of a 2-dimensional vector field Examples:

  25. General form of a 2-dimensional vector field Examples:

  26. General form of a 2-dimensional vector field Examples: QUESTION: How can we visualize 2-dimensional vector fields?

  27. General form of a 2-dimensional vector field Examples: Question: How can we visualize 2-dimensional vector fields? Answer: Draw a few representative vectors.

  28. Example:

  29. Example:

  30. We will turn over sketching vector fields in 3-space to MAPLE.

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