   Chapter 16.1, Problem 36E

Chapter
Section
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 F(x,y)=i+xj , several approximated flow lines and flow line equations.

Explanation

Given data:

F(x,y)=i+xj

Formula used:

Consider a two-dimensional vector F=x,y .

Write the expression for length of the two dimensional vector.

|F(x,y)|=x2+y2 (1)

Consider the velocity field F(x,y) .

F(x,y)=i+xj=1,x

Find the length of F(x,y) using equation (1).

|F(x,y)|=(1)2+(x)2=1+x2

Consider a certain interval of x as (2,2) and y as (2,2) to plot F(x,y) .

The estimated values of |F(x,y)| and F(x,y) for different values of x and y are shown in Table 1.

Table 1

 Quadrant (x,y) |F(x,y)|=1+x2 F(x,y)=〈1,x〉 I (0,0) 1 〈1,0〉 (1,0) 2 〈1,1〉 (2,0) 5 〈1,2〉 (0,1) 1 〈1,0〉 (1,1) 2 〈1,1〉 (0,2) 1 〈1,0〉 (

(b)

To determine

To deduce: The differential equation dydx=x .

(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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