compute the work done by the force F?

Question

compute the work done by the force F(x,y)=, C is the square from (0,0) to (1,0) to (1,1) to (0,1) to (0,0)

Answer 1

You need to know whether the force is conservative, such as in gravity. In which case the total work done is zero, as you've returned to where you started. Or if proportional to distance travelled, in which case you have travelled 4 units and so the work done would be F*4.

Answer 2

So, what's the problem?

What do you mean by ? That usually means inner product.

If so, then you need to represent F as a vector, take the inner product with the differential vector along the square and integrate.

So, for the (0,0) to (1,0) leg

F(x,y) = 0! no work

(1,0) to (1,1)

F(x,y) = x*y[ cos (arctan(y/x)) + sin (arctan(y/x))]

for the x and y components. Now dot with dr = dy, you'll only get the y component of force. This makes sense because moving a force perpendicular to it's direction does no work

dw = x*y[sin(arctan(y/x))]dy

now integrate from (1,0) to (1,1).

Do the same for the other sides.