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# MA 242.003 - PowerPoint PPT Presentation

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 .

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• Day 58 – April 9, 2013

The material we will cover before test #4 is:

• Section 10.5: Parametric surfaces

• Section 10.5: Parametric surfaces

• Pages 777-778: Tangent planes to parametric surfaces

• Section 10.5: Parametric surfaces

• Pages 777-778: Tangent planes to parametric surfaces

• Section 12.6: Surface area of parametric surfaces

• 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:

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.

Parameterized curves are 1-dimensional.

Parameterized curves are 1-dimensional.

We generalize to parameterized surfaces, which are 2-dimensional.

1. The functions

1. The functions

2. The domain D

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

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.

that lies above the unit square x = 0..1, y = 0..1 in the first octant.

that lies above the unit square x = 0..1, y = 0..1 in the first octant.

that lies above the unit square x = 0..1, y = 0..1 in the first octant.

that lies above the unit square x = 0..1, y = 0..1 in the first octant.

that lies above the unit square x = 0..1, y = 0..1 in the first octant.

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)>

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

r(u,v) = (u,g(u,v),v)

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

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)