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Unformatted text preview: Section 17.6 Parametric Surfaces Parametric Equations Defining Surfaces In Chapter 14, we discussed how curves could be represented in space through the use of parametric equations in one variable. In this section, we consider the same idea for surfaces. 1. Parametric Surfaces Suppose that vector r ( u, v ) = x ( u, v ) vector i + y ( u, v ) vector j + z ( u, v ) vector k is a vector valued function defined on a region D in 3-space. For each value of ( u, v ), we associate the point ( x ( u, v ) , y ( u, v ) , z ( u, v )) to the vector vector r ( u, v ). Then as ( u, v ) vary over D , the vector function vector r ( t ) traces out a surface S called the parametric surface with parametric equations vector r ( u, v ). We illustrate. Example 1.1. Identify the surface with parametric equations vector r ( u, v ) = u cos ( v ) vector i + u sin( v ) vector j + u 2 vector k. Since x = u cos( v ), y = u sin( v ) and z = u 2 , at any point on this surface we have x 2 + y 2 = u 2 = z . This is the equation for a parabolic bowl centered on the z-axis with vertex at the origin....
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