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MA 242.003

MA 242.003 . Day 58 – April 9, 2013. MA 242.003 . The material we will cover before test #4 is:. MA 242.003 . Section 10.5: Parametric surfaces. MA 242.003 . Section 10.5: Parametric surfaces Pages 777-778: Tangent planes to parametric surfaces. MA 242.003 .

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MA 242.003

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  1. MA 242.003 • Day 58 – April 9, 2013

  2. MA 242.003 The material we will cover before test #4 is:

  3. MA 242.003 • Section 10.5: Parametric surfaces

  4. MA 242.003 • Section 10.5: Parametric surfaces • Pages 777-778: Tangent planes to parametric surfaces

  5. MA 242.003 • Section 10.5: Parametric surfaces • Pages 777-778: Tangent planes to parametric surfaces • Section 12.6: Surface area of parametric surfaces

  6. MA 242.003 • Section 10.5: Parametric surfaces • Pages 777-778: Tangent planes to parametric surfaces • Section 12.6: Surface area of parametric surfaces • Section 13.6: Surface integrals

  7. Recall the following from chapter 10 on parametric CURVES:

  8. Recall the following from chapter 10 on parametric CURVES:

  9. Recall the following from chapter 10 on parametric CURVES: Example:

  10. Space curves DEFINITION: A space curve is the set of points given by the ENDPOINTS of the Vector-valued function when the vector is in position vector representation.

  11. My standard picture of a curve:

  12. My standard picture of a curve: Parameterized curves are 1-dimensional.

  13. My standard picture of a curve: Parameterized curves are 1-dimensional. We generalize to parameterized surfaces, which are 2-dimensional.

  14. NOTE: To specify a parametric surface you must write down: 1. The functions

  15. NOTE: To specify a parametric surface you must write down: 1. The functions 2. The domain D

  16. We will work with two types of surfaces:

  17. We will work with two types of surfaces: Type 1: Surfaces that are graphs of functions of two variables

  18. We will work with two types of surfaces: Type 1: Surfaces that are graphs of functions of two variables Type 2: Surfaces that are NOTgraphs of functions of two variables

  19. First consider Type 1 surfaces that are graphs of functions of two variables.

  20. An example: Let S be the surface that is the portion of that lies above the unit square x = 0..1, y = 0..1 in the first octant.

  21. An example: Let S be the surface that is the portion of that lies above the unit square x = 0..1, y = 0..1 in the first octant.

  22. An example: Let S be the surface that is the portion of that lies above the unit square x = 0..1, y = 0..1 in the first octant.

  23. An example: Let S be the surface that is the portion of that lies above the unit square x = 0..1, y = 0..1 in the first octant.

  24. An example: Let S be the surface that is the portion of that lies above the unit square x = 0..1, y = 0..1 in the first octant.

  25. An example: Let S be the surface that is the portion of that lies above the unit square x = 0..1, y = 0..1 in the first octant. General Rule If S is given by z = f(x,y) then r(u,v) = <u, v, f(u,v)>

  26. General Rule: If S is given by y = g(x,z) then r(u,v) = (u,g(u,v),v)

  27. General Rule: If S is given by x = h(y,z) then r(u,v) = (h(u,v),u,v)

  28. Consider next Type 2 surfaces that are NOT graphs of functions of two variables.

  29. Consider next Type 2 surfaces that are NOT graphs of functions of two variables. Spheres

  30. Consider next Type 2 surfaces that are NOT graphs of functions of two variables. Spheres Cylinders

  31. 2. Transformation Equations

  32. Introduce cylindrical coordinates centered on the y-axis

  33. Each parametric surface has a u-v COORDINATE GRID on the surface!

  34. Each parametric surface has a u-v COORDINATE GRID on the surface!

  35. Each parametric surface has a u-v COORDINATE GRID on the surface! r(u,v)

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