ma 242.003
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MA 242.003. Day 60 – April 11, 2013. MA 242.003. The material we will cover before test #4 is:. MA 242.003. Day 60 – April 11, 2013 Section 10.5: Parametric surfaces Pages 777-778: Tangent planes to parametric surfaces Section 12.6: Surface area of parametric surfaces - PowerPoint PPT PresentationTRANSCRIPT
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MA 242.003
• Day 60 – April 11, 2013
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MA 242.003The material we will cover before test #4 is:
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MA 242.003
• Day 60 – April 11, 2013• 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
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NOTE: To specify a parametric surface you must write down:1. The functions
2. The domain D
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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
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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.
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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.
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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)>
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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)
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Consider next Type 2 surfaces that are NOT graphs of functions of two variables.
Spheres
Cylinders
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Each parametric surface as a u-v COORDINATE GRID on the surface!
r(u,v)
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More generally, let S be the parametric surface traced out by the vector-valued function
as u and v vary over the domain D.
Pages 777-778: Tangent planes to parametric surfaces
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Section 12.6: Surface area of parametric surfaces
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Section 12.6: Surface area of parametric surfaces
As an application of double integration, we compute the surface area of a parameterized surface S.
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Section 12.6: Surface area of parametric surfaces
As an application of double integration, we compute the surface area of a parameterized surface S.
First recall the definition of a double integral over a rectangle.
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Section 12.6: Surface area of parametric surfaces
Goal: To compute the surface area of a parametric surface given by
with u and v in domain D in the uv-plane.
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Section 12.6: Surface area of parametric surfaces
Goal: To compute the surface area of a parametric surface given by
with u and v in domain D in the uv-plane.
1. Partition the region D, which also partitions the surface S
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Section 12.6: Surface area of parametric surfaces
Goal: To compute the surface area of a parametric surface given by
with u and v in domain D in the uv-plane.
1. Partition the region D, which also partitions the surface S
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1. Partition the region D, which also partitions the surface S
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1. Partition the region D, which also partitions the surface S
Now let us approximate the area of the patch .
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The EDGES of the patch can be approximated by vectors.
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The EDGES of the patch can be approximated by vectors.
In turn these vectors can be approximated by the vectors
and
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So we approximate by the Parallelogram determined by
and
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So we approximate by the Parallelogram determined by
and
The area of this parallelogram is
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So we approximate by the Parallelogram determined by
and
The area of this parallelogram is
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So we approximate by the Parallelogram determined by
and
The area of this parallelogram is
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Now find the surface area.
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(continuation of example)
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(continuation of example)
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(continuation of example)
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