de Rham cohomology

de Rham cohomology is a mathematical tool used to study the shape and structure of smooth spaces, like surfaces and manifolds. It analyzes how different smooth functions and differential forms behave on these spaces, capturing features such as holes or voids. By examining the properties of these forms, de Rham cohomology helps identify and classify the intrinsic "holes" or topological features that can't be removed by smooth deformation. Essentially, it provides a bridge between calculus and topology, offering a systematic way to understand the underlying structure of complex shapes through differential calculus.