Chapter 16.1, Problem 36E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

(a) Sketch the vector field **F**(*x*, *y*) = **i** + *x* **j** and then sketch some flow lines. What shape do these flow lines appear to have?

(b) If parametric equations of the flow lines are *x = x*(*t*), *y = y*(*t*), what differential equations do these functions satisfy? Deduce that *dy*/*dx = x.*

(c) If a particle starts at the origin in the velocity field given by **F**, find an equation of the path it follows.

**(a)**

To determine

**To sketch:** The vector field

Explanation

**Given data:**

**Formula used:**

Consider a two-dimensional vector

Write the expression for length of the two dimensional vector.

Consider the velocity field

Find the length of

Consider a certain interval of *x* as
*y* as

The estimated values of
*x* and *y* are shown in Table 1.

Table 1

Quadrant | |||

I | |||

**(b)**

To determine

**To deduce:** The differential equation

** (c)**

To determine

**To find:** An equation of the path if a particle starts at the origin in the velocity field.

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