Question

1130 chapter 16 vector calculus
13 - 18 match the vector fields f with the plots labeled i - vi. give reasons for your choices.

1. f(x,y)=(x, - y)
2. f(x,y)=(y,x - y)
3. f(x,y)=(y,y + 2)
4. f(x,y)=(y,2x)
5. f(x,y)=(sin y,cos x)
6. f(x,y)=(cos(x + y),x)

Explanation:

Step1: Analyze F(x,y)=(x, -y)

At (x,0) vectors are horizontal (y - component is 0), at (0,y) vectors are vertical. As x increases, horizontal component of vector increases, as y increases (in magnitude), vertical - component of vector increases in magnitude but points downwards. This matches plot II.

Step2: Analyze F(x,y)=(y,x - y)

When x = 0, F=(y, - y). When y = 0, F=(0,x). The vectors have a more complex pattern. At points on the y - axis (x = 0), vectors point down - left or up - right depending on the sign of y. This matches plot III.

Step3: Analyze F(x,y)=(y,y + 2)

The x - component of the vector is y and the y - component is y+2. When y = 0, the vector is (0,2). As y increases, both components increase. Vectors are parallel to each other in horizontal strips. This matches plot V.

Step4: Analyze F(x,y)=(y,2x)

When x = 0, vectors are vertical (x - component is 0). When y = 0, vectors are horizontal. The magnitude of the y - component depends on y and the magnitude of the x - component depends on x. This matches plot IV.

Step5: Analyze F(x,y)=(sin y,cos x)

Since - 1≤sin y≤1 and - 1≤cos x≤1, the vectors have bounded x and y components. The vectors will have a more 'wavy' or periodic - like pattern. This matches plot VI.

Step6: Analyze F(x,y)=(cos(x + y),x)

The x - component of the vector is x and the y - component is cos(x + y). The x - component varies linearly with x and the y - component is periodic in x + y. This matches plot I.

Answer:

1. F(x,y)=(x, -y) matches II.
2. F(x,y)=(y,x - y) matches III.
3. F(x,y)=(y,y + 2) matches V.
4. F(x,y)=(y,2x) matches IV.
5. F(x,y)=(sin y,cos x) matches VI.
6. F(x,y)=(cos(x + y),x) matches I.