   Chapter 16.6, Problem 4E

Chapter
Section
Textbook Problem

Identify the surface with the given vector equation.4. r(u, v) = u2 i + u cos v j + u sin v k

To determine

To identify: The surface with the vector equation r(u,v)=u2i+ucosvj+usinvk .

Explanation

Given data:

The vector equation is given as follows.

r(u,v)=u2i+ucosvj+usinvk

Formula used:

Write the expression for circular paraboloid that symmetric to the x-axis.

x=y2+z2 (1)

Write the vector equation as follows.

r(u,v)=u2i+ucosvj+usinvk

Write the parametric equations from the vector equation as follows.

x=u2,y=ucosv,z=usinv

Take square on both sides of the symmetric expression y=ucosv .

y2=(ucosv)2=u2cos2v

Take square on both sides of the symmetric expression z=usinv .

z2=(usinv)2=u2sin2v

Add the expressions y2 and z2 as follows

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