I've never had any experience with differential forms before, so I'm trying to work through a couple of examples to see if I understand what's going on. I think I understand what I've been doing so far, but I'd like a little bit of reassurance or correction before I go too much further.
First, consider the differential form $\alpha = x dy-\frac{1}{2}(x^2+y^2)dt$ in $\mathbb{R}^3=\{(x,y,t)\}$. Then $ \begin{align*} d\alpha &= dx\wedge dy - \frac{1}{2}(2x dx+2y dy)dt \\&= dx\wedge dy- x dx\wedge dt- y dy\wedge dt \end{align*}$ Then we also have that $ dt\wedge d\alpha = dt\wedge dx\wedge dy = dx\wedge dy\wedge dt \qquad \text{by anticommutativity} $ and $ dx\wedge d\alpha = -y dx\wedge dy\wedge dt $ Are these computations correct?
Next, I'd like to know if $\alpha$ is a differential of a globally defined smooth function on $\mathbb{R}^3$. I'm not entirely sure what this means, but I think the question is asking if there is some globally defined smooth function on $\mathbb{R}^3$ whose total derivative is the differential $\alpha$. A globally defined smooth function is one that has no discontinuities, is infinitely differentiable, and every derivative is a continuous function. It seems like it should be relatively easy to find such a function, but I haven't managed to do so. Is there some sort of criterion that makes this question easier?