Potential Function from a simple conservative force

[K29]

Homework Statement

Find the potential functions for these conservative forces:

(1)F=xi+yj

(2)F=yi+xj

Homework Equations

F=-$$\nabla$$V (Force = -del (Pot.energy))

The Attempt at a Solution

So, I'm guessing to get V I just need to integrate F. For the first equation that gets me the right answer hurray!

However I think my method is wrong(or my general understanding the opposite of partial derivatives). Doing the same thing to eqn 2 leaves me with a slight problem.
-($$\int$$y dx +$$\int$$x dy)leaves me with -2xy+C. The correct answer should be -xy+C

(I checked and it is indeed a conservative force)

Please enlighten me
Thanks in advance

 

[ehild]

For a conservative force, the work between two points is independent on the connecting path. Start to integrate from point (0,0) to point P(X,Y).
You can follow the line along the x-axis from x=0 to x=X and then a straigth line parallel to the y-axis from (X,0) to (X,Y). The work done is

$$V(0,0)-V(X,Y)=\int_{(0,0)}^{(X,Y)}{ydx+xdy}=\int_{(0,0)}^{(0,X)}{ydx}+\int_{(X,0)}^{(X,Y)}{xdy}$$

As y=0 along the first line, and x=X along the second one, the first integral is 0, the second one is XY.

ehild

 

[K29]

solved. thanks