Given a force \( \vec{F}=y \hat{\mathbf{x}}-x \hat{\mathbf{y}} \), compute the path integral \( \int \vec{F} \cdot \mathrm{d} \vec{r} \) from \( (0,0) \) to \( (1,1) \) along the paths given by (a) The line from \( (0,0) \) to \( (0,1) \) then the line from \( (0,1) \) to \( (1,1) \). (b) The curve \( y=x^{3} \). (c) The curve \( y=\sqrt{x} \).
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