58
MA 242.003 • Day 58 – April 9, 2013
MA 242.003
• Day 58 – April 9, 2013
MA 242.003The 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
• Section 10.5: Parametric surfaces• Pages 777-778: Tangent planes to parametric
surfaces• Section 12.6: Surface area of parametric surfaces
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
Recall the following from chapter 10 on parametric CURVES:
Recall the following from chapter 10 on parametric CURVES:
Recall the following from chapter 10 on parametric CURVES:
Example:
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.
My standard picture of a curve:
My standard picture of a curve:
Parameterized curves are 1-dimensional.
My standard picture of a curve:
Parameterized curves are 1-dimensional.We generalize to parameterized surfaces, which are 2-dimensional.
NOTE: To specify a parametric surface you must write down:1. The functions
NOTE: To specify a parametric surface you must write down:1. The functions
2. The domain D
We will work with two types of surfaces:
We will work with two types of surfaces:
Type 1: Surfaces that are graphs of functions of two variables
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
First consider Type 1 surfaces that are graphs of functions of two variables.
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.
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.
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.
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.
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.
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)>
General Rule:
If S is given by y = g(x,z) then
r(u,v) = (u,g(u,v),v)
General Rule:
If S is given by x = h(y,z) then
r(u,v) = (h(u,v),u,v)
Consider next Type 2 surfaces that are NOT graphs of functions of two variables.
Consider next Type 2 surfaces that are NOT graphs of functions of two variables.
Spheres
Consider next Type 2 surfaces that are NOT graphs of functions of two variables.
Spheres
Cylinders
2. Transformation Equations
Introduce cylindrical coordinates centered on the y-axis
Each parametric surface has a u-v COORDINATE GRID on the surface!
Each parametric surface has a u-v COORDINATE GRID on the surface!
Each parametric surface has a u-v COORDINATE GRID on the surface!
r(u,v)
