Chapter 16.6, Problem 14E

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8th Edition

James Stewart

ISBN: 9781285741550

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.

**14. r**(*u*, *v*) = *uv*^{2} **i** + *u*^{2}*v* **j** + (*u*^{2} − *v*^{2}) **k**

To determine

**To match:** The equation

Explanation

**Given data:**

The given vector equation as follows.

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

Write the vector equation as follows.

Write the parametric equations from the vector equation as follows.

**Determination of grid curves when**
** is constant:**

Consider the parameter

Substitute

From the expression

Substitute

As

Therefore, the expression

The expression represents the equation of parabola with the axis as *x*-axis.

Find the

Substitute

From equation (2) and (3), each grid curve corresponds to
*xy*-plane is a parabola with the axis as *x*-axis

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