I am studying Vector Calculus and Poincare's lemma was mentioned in the class notes.
Poincaré Lemma: If $U \subset \Bbb{R}^n$ is an open star-shaped set, then for every $k=\{0,…,n\}$ every CLOSED k-form in $U$ is EXACT.
I want to know if I understand the concept or the significance of the result applied to vector fields: basically, if we have a vector field defined in a star-shaped domain and its divergence is zero, then the vector field has a vector potential?
Now to my second question. I've seen some people use contractible domain in the definition instead of star-shaped. I know that a star-shaped domain is a contractible domain but a contractible domain isn't necessarily a star-shaped domain. Is it necessary for that the domain is star-shaped or it holds if the domain is just contractible? (a lot of sources I have read/asked always mention that the domain needs to be able to be continuously contracted to a point).
Best Answer
Many texts assume a star-shaped domain so that one can write down an explicit integral along rays from the "star point" to give a primitive of the closed form. But all one needs is a contractible set in order for the Poincaré lemma to be valid, and there is, in fact, a general formula obtained by integrating over the fibers of the null-homotopy.