Chapter 16.1, Problem 35E

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8th Edition

James Stewart

ISBN: 9781305266643

Textbook Problem

**The flow lines** (or **streamlines**) of a vector field are the paths followed by a particle whose velocity field is the given vector field. Thus the vectors in a vector field are tangent to the flow lines.

(a) Use a sketch of the vector field **F**(*x*, *y*) = *x* **i** − *y* **j** to draw some flow lines. From your sketches, can you guess the equations of the flow lines?

(b) If parametric equations of a flow line are *x* = *x*(*t*), *y* = *y*(*t*), explain why these functions satisfy the differential equations *dx*/*dt* = *x* and *dy*/*dt* = −*y.* Then solve the differential equations to find an equation of the flow line that passes through the point (1, 1).

**(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

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Quadrant | |||

I | |||

**(b)**

To determine

**To solve:** The differential equations to find the flow line that passes through the point

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