   Chapter 16.6, Problem 16E

Chapter
Section
Textbook Problem

Match the equations with the graphs labeled I–VI and give reasons for your answers. Determine which families of grid curves have u constant and which have v constant.16. x = (1 − u)(3 + cos v) cos 4πu, y = (1 − u)(3 + cos v) sin 4πu, z = 3u + (1 − u) sin v To determine

To match: The parametric equations x=(1u)(3+cosv)cos4πu,y=(1u)(3+cosv)sin4πu,z=3u+(1u)sinv with the graphs labeled I-VI in the textbook and to explain the reason for the combination.

Explanation

Given data:

Write the given parametric equations as follows.

x=(1u)(3+cosv)cos4πu,y=(1u)(3+cosv)sin4πu,z=3u+(1u)sinv

The graphs are labeled in the textbook from I to VI.

Determination of grid curves when u is constant:

Consider the parameter u=0 .

Substitute 0 for u in the parametric equations as follows.

x=(10)(3+cosv)cos4π(0),y=(10)(3+cosv)sin4π(0),z=3(0)+(10)sinvx=(3+cosv)(1),y=(3+cosv)(0),z=0+sinvx=3+cosv,y=0,z=sinv

From the expression x=3+cosv,y=0,z=sinv , it is clear that the equation represents the circle with radius 1 in the xz-plane centered at the point (3,0,0) .

Consider the parameter u=12 .

Substitute 12 for u in the parametric equations,

x=(112)(3+cosv)cos4π(12),y=(112)(3+cosv)sin4π(12),z=3(12)+(112)sinvx=(12)(3+cosv)(1),

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