   Chapter 16.6, Problem 15E

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.15. r(u, v) = (u3 − u) i + v2 j + u2 k To determine

To match: The equation r(u,v)=(u3u)i+v2j+u2k 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)=(u3u)i+v2j+u2k

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

Write the vector equation as follows.

r(u,v)=(u3u)i+v2j+u2k

Write the parametric equations from the vector equation as follows.

x=u3u,y=v2,z=u2 (1)

Determination of grid curves when u is constant:

If the parameter u is held constant in the equation (1), the x- and z-parameters become constant and only the parameter y is a variable.

Therefore, each grid curve is contained in a straight line parallel to the y-axis when the parameter u is constant

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