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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9.

5. F ( x , y ) = y i + x j x 2 + y 2

To determine

To sketch: The vector field F for equation F(x,y)=yi+xjx2+y2 .

Explanation

Given data:

F(x,y)=yi+xjx2+y2

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 given vector field.

F(x,y)=yi+xjx2+y2=yx2+y2i+xx2+y2j=yx2+y2,xx2+y2

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

|F(x,y)|=(yx2+y2)2+(xx2+y2)2=(y2x2+y2)+(x2x2+y2)=y2+x2x2+y2=1

|F(x,y)|=1

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

(x,y) |F(x,y)|=1 F(x,y)=yx2+y2,xx2+y2
(0,0) 1 0,0
(1,0) 1 0,1
(

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