Find the circulation done by the velocity field V(x, y)= -y/[(x+1)^2 +4y^2]i + (x+1)/[(x+1)^2 +4y^2]j around the positively oriented square with vertices (2, 0), (0, 2), (−2, 0), and (0, −2).

The circulation of V around the given path (call it C) is simply the line integral of V along C. Since C is a closed curve, we can use Green's theorem.

where int(C) is the interior or region bounded by C.

We have

so the double integral and hence the circulation is zero.

question 1 is 8, question 2 is 9, question 3 is 24

step-by-step explanation:

now, the numbers they give you at the top are the key. from there, you just have to plug and chug! let's try 1, 2, and 3!

key: a=3, b=5, c=6

1. a+5

remember, a=3, so we plug that in to get 3+5=8

2. 15-c

c=6, so we plug that in to get 15-6=9

3. 4b

b=6, so we plug it in to get 4*6=24

time to try the rest on your own! if you put the answers below in a comment, i'll check them for you!

d 25 pi/6

step-by-step explanation:

the diameter is 10 inches

the radius is 1/2 of the diameter or 5 inches

the area of the circle is

a = pi r^2

  = pi 5^2

  = 25 pi

the slice is 60 degrees of the entire circle.

60 degrees out of 360   60/360 = 1/6

the area of the slice is 1/6 of the circle

a slice = 1/6 a circle

a slice = 1/6 * 25 pi