MA 242.003

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)