Problem 3.16. Find a vector field \(\vec{A}(\vec{r})\) in \(\mathbb{R}^3\) such that:
\(\vec{\nabla} \times \vec{A} = y^2 \cos(y)e^{-y^2}\hat{i} + x \sin(x)e^{-x^2}\hat{j} = (y^2 \cos(y)e^{-y^2}, x \sin(x)e^{-x^2}, 0)\).
Solution. Using Thm. 3.5, a possible solution is
\(\vec{A}(\vec{r}) = z^*(\vec{\nabla} \times \vec{A}) = (zx \sin(x)e^{-x^2}, -zy^2 \cos(y)e^{-y^2}, 0).\)