Finding the circulation of a vector field

[polaris90]

Homework Statement

Can someone guide me through solving a problem involving the circulation of a vector field?
The question is as stated
for the vector field [itex] E = (xy)X^ - (x^2 + 2y^2)Y^ [/itex], where the letters next to the parenthesis with the hat mean they x y vector component. I need to find the circulation of that vector field given by the close path of a triangle going from (0,0) to (0,1) to (1,1) and back to (0,0)

Homework Equations

circulation = ∫Bdl

The Attempt at a Solution

I tried taking the partial derivatives with respect to x and y and then added the results. Then I took the double integral from 0 to 1 for the given result. The answer in the back of the book says -1. The only examples I see everywhere is the one with the circle which is very simple and only has one path. I understand I can add the integration of the individual paths, so how can I do the last part goinf from (1,1) to (0,0) when the vector changes for both variables?

 

[LCKurtz]

Homework Statement

Can someone guide me through solving a problem involving the circulation of a vector field?
The question is as stated
for the vector field [itex] E = (xy)X^ - (x^2 + 2y^2)Y^ [/itex], where the letters next to the parenthesis with the hat mean they x y vector component. I need to find the circulation of that vector field given by the close path of a triangle going from (0,0) to (0,1) to (1,1) and back to (0,0)

Homework Equations

circulation = ∫Bdl

The Attempt at a Solution

I tried taking the partial derivatives with respect to x and y and then added the results. Then I took the double integral from 0 to 1 for the given result. The answer in the back of the book says -1. The only examples I see everywhere is the one with the circle which is very simple and only has one path. I understand I can add the integration of the individual paths, so how can I do the last part goinf from (1,1) to (0,0) when the vector changes for both variables?

You need to show your work. Among other possible errors, integrating both x and y from 0 to 1 would describe a square, not a triangle. Are you using Green's theorem? That would give an integrand of ##Q_x - P_y## not their sum.

 

[polaris90]

Thank you for your reply. I didn't know what I was doing, but I used Green's theorem as you said and helped me solve the problem. I integrated with respect to x from 0 to 1 and with respect to y from 0 to x and I obtained the right answer.