(Solved): 4. Let \[ \mathbf{F}(x, y)=\langle y \sin y+1, x \sin y+x y \cos y-6 y\rangle . \] (i) Show that \( ...

4. Let \[ \mathbf{F}(x, y)=\langle y \sin y+1, x \sin y+x y \cos y-6 y\rangle . \] (i) Show that \( F \) is conservative. (ii) Find a potential function for \( F \). (iii) Use the potential function and the Fundamental Theorem of Line Integrals to evaluate \[ \int_{C} \mathbf{F} \cdot d \mathbf{r} \] where \( C \) is any curve joining \( (0,1) \) to \( (1,0) \).