ma 242.003 day 60 – april 11, 2013. ma 242.003 the material we will cover before test #4 is:
TRANSCRIPT
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
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)
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
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
Now let us approximate the area of the patch .
The EDGES of the patch can be approximated by vectors.
In turn these vectors can be approximated by the vectors
and