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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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BuyFindarrow_forward

Multivariable Calculus

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 iy 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 F(x,y)=xiyj , several approximated flow lines and flow line equations.

Explanation

Given data:

F(x,y)=xiyj

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)=xiyj=x,y

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

|F(x,y)|=(x)2+(y)2=x2+y2

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)|=x2+y2 F(x,y)=x,y
I (0,0) 0 0,0
(1,0) 1 1,0
(2,0) 2 2,0
(0,1) 1 0,1
(1,1) 2 1,1
(0,2) 2 0,2
(2,2) 22 2,2

(b)

To determine

To solve: The differential equations to find the flow line that passes through the point (1,1) .

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