MA 242.003

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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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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

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:

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.

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

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)>

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.

Spheres

Consider next Type 2 surfaces that are NOT graphs of functions of two variables.

Spheres

Cylinders

r(u,v)