MA 242.003
description
Transcript of MA 242.003
![Page 1: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/1.jpg)
MA 242.003
• Day 58 – April 9, 2013
![Page 2: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/2.jpg)
MA 242.003The material we will cover before test #4 is:
![Page 3: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/3.jpg)
MA 242.003
• Section 10.5: Parametric surfaces
![Page 4: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/4.jpg)
MA 242.003
• Section 10.5: Parametric surfaces• Pages 777-778: Tangent planes to parametric
surfaces
![Page 5: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/5.jpg)
MA 242.003
• Section 10.5: Parametric surfaces• Pages 777-778: Tangent planes to parametric
surfaces• Section 12.6: Surface area of parametric surfaces
![Page 6: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/6.jpg)
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
![Page 7: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/7.jpg)
Recall the following from chapter 10 on parametric CURVES:
![Page 8: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/8.jpg)
Recall the following from chapter 10 on parametric CURVES:
![Page 9: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/9.jpg)
Recall the following from chapter 10 on parametric CURVES:
Example:
![Page 10: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/10.jpg)
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.
![Page 11: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/11.jpg)
![Page 12: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/12.jpg)
![Page 13: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/13.jpg)
My standard picture of a curve:
![Page 14: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/14.jpg)
My standard picture of a curve:
Parameterized curves are 1-dimensional.
![Page 15: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/15.jpg)
My standard picture of a curve:
Parameterized curves are 1-dimensional.We generalize to parameterized surfaces, which are 2-dimensional.
![Page 16: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/16.jpg)
![Page 17: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/17.jpg)
![Page 18: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/18.jpg)
![Page 19: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/19.jpg)
NOTE: To specify a parametric surface you must write down:1. The functions
![Page 20: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/20.jpg)
NOTE: To specify a parametric surface you must write down:1. The functions
2. The domain D
![Page 21: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/21.jpg)
We will work with two types of surfaces:
![Page 22: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/22.jpg)
We will work with two types of surfaces:
Type 1: Surfaces that are graphs of functions of two variables
![Page 23: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/23.jpg)
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
![Page 24: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/24.jpg)
First consider Type 1 surfaces that are graphs of functions of two variables.
![Page 25: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/25.jpg)
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.
![Page 26: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/26.jpg)
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.
![Page 27: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/27.jpg)
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.
![Page 28: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/28.jpg)
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.
![Page 29: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/29.jpg)
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.
![Page 30: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/30.jpg)
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 RuleIf S is given by z = f(x,y) then
r(u,v) = <u, v, f(u,v)>
![Page 31: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/31.jpg)
![Page 32: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/32.jpg)
General Rule:
If S is given by y = g(x,z) then
r(u,v) = (u,g(u,v),v)
![Page 33: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/33.jpg)
![Page 34: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/34.jpg)
General Rule:
If S is given by x = h(y,z) then
r(u,v) = (h(u,v),u,v)
![Page 35: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/35.jpg)
Consider next Type 2 surfaces that are NOT graphs of functions of two variables.
![Page 36: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/36.jpg)
Consider next Type 2 surfaces that are NOT graphs of functions of two variables.
Spheres
![Page 37: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/37.jpg)
Consider next Type 2 surfaces that are NOT graphs of functions of two variables.
Spheres
Cylinders
![Page 38: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/38.jpg)
![Page 39: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/39.jpg)
2. Transformation Equations
![Page 40: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/40.jpg)
![Page 41: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/41.jpg)
![Page 42: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/42.jpg)
Introduce cylindrical coordinates centered on the y-axis
![Page 43: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/43.jpg)
Each parametric surface has a u-v COORDINATE GRID on the surface!
![Page 44: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/44.jpg)
Each parametric surface has a u-v COORDINATE GRID on the surface!
![Page 45: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/45.jpg)
Each parametric surface has a u-v COORDINATE GRID on the surface!
r(u,v)
![Page 46: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/46.jpg)
![Page 47: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/47.jpg)
![Page 48: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/48.jpg)
![Page 49: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/49.jpg)
![Page 50: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/50.jpg)
![Page 51: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/51.jpg)
![Page 52: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/52.jpg)
![Page 53: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/53.jpg)
![Page 54: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/54.jpg)
![Page 55: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/55.jpg)
![Page 56: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/56.jpg)
![Page 57: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/57.jpg)
![Page 58: MA 242.003](https://reader035.fdocuments.net/reader035/viewer/2022062410/5681611f550346895dd07928/html5/thumbnails/58.jpg)