- #1

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[itex]\vec{F} = y\hat{x} + x\hat{y}[/itex]

This force is conservative ([itex]\nabla \times \vec{F} = 0[/itex]).

So I integrate it to find the potential energy:

[itex]U = -\int \vec{F} \bullet d\vec{s}[/itex]

[itex]U = -\int y \delta x - \int x \delta y[/itex]

[itex]U = -yx - xy[/itex]

[itex]U = -2xy[/itex]

Ignoring the arbitrary constant because it can be set to 0.

And now to do what should be the reverse operation:

[itex]\vec{F} = -\nabla U[/itex]

[itex]\vec{F} = -\hat{x}\frac{\delta U}{\delta x} - \hat{y}\frac{\delta U}{\delta y}[/itex]

[itex]\vec{F} = 2y\hat{x} + 2x\hat{y}[/itex]

So somehow, in the process of integrating and then differentiating, the force doubled. I'm not sure where the error is here but I'd very much like to get rid of it. Can anyone point me in the right direction here please?