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:

9. 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.

13. My standard picture of a curve:

14. Parameterized curves are 1-dimensional.

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

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

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

21. We will work with two types of surfaces:

22. Type 1: Surfaces that are graphs of functions of two variables

23. We will work with two types of surfaces: Type 1: Surfaces that are graphs of functions of two variables Type 2: Surfaces that are NOT graphs of functions of two variables

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

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.

26. 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.

27. 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.

28. 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.

29. 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.

30. 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) =

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

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

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

36. Spheres

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

39. 2. Transformation Equations

42. Introduce cylindrical coordinates centered on the y-axis

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

45. r(u,v)