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MA 242.003

- Day 58 – April 9, 2013

MA 242.003

The 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:

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:

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

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 NOTgraphs 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 Rule

If S is given by z = f(x,y) then

r(u,v) = <u, v, f(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

2. Transformation Equations

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)

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