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Overview of Parametric Surfaces and Tangent Planes in Calculus MA.242.003

This review session for MA.242.003 covers key concepts related to parametric surfaces before Test #4. We will discuss sections 10.5, 12.6, and 13.6, focusing on tangent planes, surface area, and surface integrals of parametric surfaces. We'll also recap important ideas from chapter 10 regarding parametric curves and differentiate between types of surfaces—those that are graphs and those that are not. Real-world examples and general rules will help illustrate these concepts in preparation for the upcoming assessment.

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Overview of Parametric Surfaces and Tangent Planes in Calculus 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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