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[tex] \vec{F} (x,y) = sin(y)\vec{i} + (xcos(y) + sin(y))\vec{j} [/tex]

I worked the problem and found out that the force was conservative and I found the potential function. Okay, I want to know why it is considered conservative if

[tex] \frac{\partial Q} {\partial x} = \frac {\partial P} {\partial y} [/tex]

is true.

where P is the scalar function sin(y) and Q is the scalar function xcos(y) + sin(y);

Now in order to work this problem this way I had to assume that the force was conservative and then imply that all conservative forces are equal to their potential function. So I had [tex] \frac{\partial f} {\partial x} = sin(y) [/tex] and [tex] \frac{\partial f} {\partial y} = xcos(y) + sin(y) [/tex]

now if I take the partial derivative of each of these with respect to the other variable then I can show that

[tex] \frac{\partial Q} {\partial x} = \frac {\partial P} {\partial y} [/tex]

by Clairaut's theorem. Well that's great that they are equal but why does that show conservatism?