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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 Presentation
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MA 242.003
• Day 60 – April 11, 2013
MA 242.003The 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• Section 13.6: Surface integrals
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:
Type 1: Surfaces that are graphs of functions of two variables
Type 2: Surfaces that are NOT 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.
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.
Spheres
Cylinders
Each parametric surface as a u-v COORDINATE GRID on the surface!
r(u,v)
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
Section 12.6: Surface area of parametric surfaces
Section 12.6: Surface area of parametric surfaces
As an application of double integration, we compute the surface area of a parameterized surface S.
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.
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.
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
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
1. Partition the region D, which also partitions the surface S
1. Partition the region D, which also partitions the surface S
Now let us approximate the area of the patch .
The EDGES of the patch can be approximated by vectors.
The EDGES of the patch can be approximated by vectors.
In turn these vectors can be approximated by the vectors
and
So we approximate by the Parallelogram determined by
and
So we approximate by the Parallelogram determined by
and
The area of this parallelogram is
So we approximate by the Parallelogram determined by
and
The area of this parallelogram is
So we approximate by the Parallelogram determined by
and
The area of this parallelogram is
Now find the surface area.
(continuation of example)
(continuation of example)
(continuation of example)