Find a $1$-form whose exterior derivative is $2x^2y^2dydz +3xz^2dzdx - 4xy^2dxdy$

Question

Find a $1$-form whose exterior derivative is $$2x^2y^2dydz +3xz^2dzdx - 4xy^2dxdy$$

An exterior calculus question. I am trying to learn some algebraic topology, and have hit a bump with some (I assume) prerequisite material.

Answer 1

Let $\omega=fdx+gdy+hdz$ be a smooth 1-form on $\mathbb{R}^3$. Then $d\omega=(\frac{\partial{h}}{\partial{y}}-\frac{\partial{g}}{\partial{z}})dy\wedge{dz}+(\frac{\partial{f}}{\partial{z}}-\frac{\partial{h}}{\partial{x}})dz\wedge{dx}+(\frac{\partial{g}}{\partial{x}}-\frac{\partial{f}}{\partial{y}})dx\wedge{dy}$.

and we can hopefully use this expression to solve for $f,g,h$.

However, as arkeet says, the $2$-form you give is not closed, hence not exact!