Parametric Surfaces We can use parametric equations to describe a curve. Because a curve is one...
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Transcript of Parametric Surfaces We can use parametric equations to describe a curve. Because a curve is one...
Parametric Surfaces We can use parametric equations to describe
a curve. Because a curve is one dimensional, we only need one
parameter. If we want to describe a surface (two dimensions) using
parametric equations, we will need two parameters. r(u,v) where the
domain refers to values in the uv-plane. Ex. Identify the surface
with vector equation r(u,v) = 2cos ui + vj + 2sin uk. Ex. Identify
the surface with vector equation r(u,v) = 5ui + (2u + v)j + v2k.
Ex. Find the rectangular equation of the surface with vector
equation r(u,v) = 2ucos vi + u2 j + 2usin vk. Ex. Find a parametric
representation of the sphere x2 + y2 + z2 = a2. Ex. Find a
parametric representation of the elliptic paraboloid z = 2x2 + y2.
For surfaces created by rotating the function y =f (x) about the
x-axis, the parametric equations would be x = uy =f (u) cos vz =f
(u) sin v These can be adapted for rotation around the y- or
z-axis. Ex. Find the parametric equation of the surface generated
by rotating z = ln y about the y-axis. Let r(u,v) = x(u,v)i +
y(u,v)j + z(u,v)k
Each of these are tangent to the surface Ex. Find the equation of
the plane tangent to the surface r(u,v) = u2i + v2j + (u + 2v)k at
the point (1,1,1). Thm. If a surface S is defined by the vector
function r(u,v), defined on the region D in the uv-plane, then the
surface area of S is Ex. Find the surface area of the sphere of
radius a. Ex. Find the area of the surface defined by z =f (x,y).
Ex. Find the area of the part of the paraboloid z = x2 + y2 that
lies under the plane z = 9. Surface Integrals Line integrals added
the values of a function at every point on a curve. Surface
integrals add the value of a 3-D function at every point on a
surface. Let S be a surface with equation z = g(x,y), and let R be
the projection of S onto the xy-plane. Ex. Evaluate , where S is
the surface z = x + y2, Ex. Evaluate , where S is the first
octant
portion of 2x + y + 2z = 6. If the surface can not be written as z
= g(x,y), then we need to parameterize it like last time.
Thm. If S can be represented parametrically by , then where D is
the domain in the uv-plane. Ex. Evaluate, where S is given by Ex.
Evaluate , where S is the unit sphere. Ex. Evaluate , where S is
the surface of the region
bounded by x2 + y2 = 1, z = 0, and z = 1 + x. Surface Integrals of
Vector Fields
Let S be an oriented surface with unit normal vector n.The surface
integral of F over S is also called the flux integral of F over S.
If F is a force causing energy to flow through our surface, the
flux integral gives the rate of flow through S. For surface defined
by z = g(x,y), the unit normal vector is Ex. Evaluate , where F =
yi + xj + zk and S is
the boundary of the solid enclosed by z = 1 x2 y2 and z = 0. For a
surface that is defined parametrically, the unit normal vector is
Ex. Find the flux of the vector field F = zi + yj + xk across the
unit sphere.