The Poincaré lemma

The Poincaré lemma states that on a smooth, "hole-free" space (called a contractible manifold), every differential form that appears to be "closed" (meaning it locally has no boundary, like a loop with no start or end) can actually be written as the "derivative" of another form (called an exact form). In simple terms, if there's no obstruction (holes) in the space, then every quantity that looks like it could be the boundary of something else truly is, allowing us to find a "potential" function for it. This is a key concept in calculus and topology related to understanding the shape of spaces.