   Chapter 16.6, Problem 13E

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.13. r(u, v) = u cos v i + u sin v j + v k To determine

To match: The equation r(u,v)=ucosvi+usinvj+vk with the graphs labeled I-VI in the textbook and to explain the reason for the combination.

Explanation

Given data:

The given vector equation as follows.

r(u,v)=ucosvi+usinvj+vk

Write the vector equation as follows.

r(u,v)=ucosvi+usinvj+vk

Write the parametric equations from the vector equation as follows.

x=ucosv,y=usinv,z=v (1)

Determination of grid curves when u is constant:

If the parameter u is held constant in the equation (1), the projection onto the xy-plane is circular with z=v .

Thus, each grid curve is a helix when the parameter u is constant

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