Exact 1- and 2-forms in $R^n$ If $D$ is a simply connected set in $R^2$ or $R^3$, then closed $1$-forms on D are exact.
This fact is suitable for the elementary vector calculus course.
I have been unable to find similarly suitable sufficient conditions on a domain $D$ in $R^n$ that closed $1$-forms are exact. 
Or that closed $2$-forms are exact.
Closed forms on a contractible set are exact.
However, "contractible" is not a suitable condition, 
as the simply connected set $R^3 - \{0\}$ is not contractible.
 A: See http://en.wikipedia.org/wiki/De_Rham_cohomology and in particular De Rham's fundamental theorem. "Closed = exact" for a domain says the De Rham cohomology group is trivial in the appropriate degree; and the condition comes down to normal topological calculations.
A: The answer "every closed form is exact if and only if the appropriate de Rham cohomology group vanishes" carries no information whatever, since you are (in effect) just giving the definition of said group. You can say that "this property is sufficiently interesting that wise men (Poincare, de Rham, et al) have built a powerful machine to figure out when a space has the property and more general versions thereof. If you are interested, please go on to study Bott-Tu [or your favorite other reference), and I will be happy to supervise you in an independent study course". I believe that giving dumbed-down reasons is counterproductive, since the students will believe that your sufficient condition is necessary and sufficient, so the next bridge they build will fall down.
