1. **Problem Statement:** Show that the vector field $\vec{F} = (x^2 + x y^2) \vec{i} + (y^2 + x^2 y) \vec{j}$ is irrotational and find its scalar potential. 2. **Recall:** A vector field $\vec{F} = P \vec{i} + Q \vec{j}$ is irrotational if its curl is zero, i.e., $$\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 0.$$ If irrotational, there exists a scalar potential function $\phi(x,y)$ such that $$\vec{F} = \nabla \phi = \frac{\partial \phi}{\partial x} \vec{i} + \frac{\partial \phi}{\partial y} \vec{j}.$$ 3. **Identify components:** $$P = x^2 + x y^2, \quad Q = y^2 + x^2 y.$$ 4. **Compute partial derivatives:** $$\frac{\partial Q}{\partial x} = \frac{\partial}{\partial x}(y^2 + x^2 y) = 0 + 2 x y = 2 x y,$$ $$\frac{\partial P}{\partial y} = \frac{\partial}{\partial y}(x^2 + x y^2) = 0 + 2 x y = 2 x y.$$ 5. **Check irrotational condition:** $$\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 2 x y - 2 x y = 0.$$ Thus, $\vec{F}$ is irrotational. 6. **Find scalar potential $\phi(x,y)$:** Since $$\frac{\partial \phi}{\partial x} = P = x^2 + x y^2,$$ integrate w.r.t. $x$: $$\phi = \int (x^2 + x y^2) dx = \frac{x^3}{3} + \frac{x^2 y^2}{2} + h(y),$$ where $h(y)$ is an arbitrary function of $y$. 7. **Determine $h(y)$ using** $$\frac{\partial \phi}{\partial y} = Q = y^2 + x^2 y.$$ Calculate $\frac{\partial \phi}{\partial y}$ from the expression: $$\frac{\partial \phi}{\partial y} = \frac{\partial}{\partial y} \left( \frac{x^3}{3} + \frac{x^2 y^2}{2} + h(y) \right) = x^2 y + h'(y).$$ Set equal to $Q$: $$x^2 y + h'(y) = y^2 + x^2 y \implies h'(y) = y^2.$$ 8. **Integrate $h'(y)$:** $$h(y) = \int y^2 dy = \frac{y^3}{3} + C,$$ where $C$ is a constant. 9. **Final scalar potential:** $$\boxed{\phi(x,y) = \frac{x^3}{3} + \frac{x^2 y^2}{2} + \frac{y^3}{3} + C}.$$