   Chapter 16.1, Problem 3E

Chapter
Section
Textbook Problem

Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9.3. F(x, y) = − 1 2 i + (y – x) j

To determine

To sketch: The vector field F for equation F(x,y)=12i+(yx)j .

Explanation

Given data:

F(x,y)=12i+(yx)j

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)=12i+(yx)j=12,(yx)

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

|F(x,y)|=(12)2+(yx)2

|F(x,y)|=14+(yx)2 (2)

Consider the line with y=x .

Substitute x for y in equation (2),

|F(x,y)|=14+(xx)2=14+0=14=12

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)| F(x,y)=〈−12,(y−x)〉 (0,0) 12 〈−12,0〉 (1,0) 12 〈−12,−1〉 (−1,0) 12 〈−12,1〉 (2,0) 12 〈

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