Find the counterclockwise circulation and outward flux of the field Fequals7 xy Bold i plus 2 y squared Bold j around and over the boundary of the region C enclosed by the curves yequalsx squared and yequalsx in the first quadrant.

The counterclockwise circulation is and the outward flux is

Step-by-step explanation:

We are given the field . A picture of the region and the path we are considering is attached. Recalll the following theorems.

Given a field of the form F(x,y)=(f(x,y),g(x,y) with f,g having continous partial derivates, C is a closed path counterclockwise oriented, R is the region enclosed by C and n is the normal vector pointing outwards of the path C. Then

(this one calculates the counterclockwise circulation)

(This one calculates the outward flux)

Then, recall that in our case f(x,y) = 7xy, g(x,y)=2y^2[/tex]. Then

.

Note that we just need to describe our region R. The region R lies between the parabola y=x^2 and the line y=x. Thus, one way to describe the region is as follows . Then, using the previous results, we get that

(circulation)

(flux)