This thesis explores and proves the theorem of De Rham using sheaf cohomology. The theorem of De Rham states that for a smooth manifold, the singular cohomology is isomorphic to the De Rham cohomology. Intuitively, this theorem states that the number of holes in a smooth manifold coincides with the failure of the fundamental theorem of calculus for closed differential forms defined on the manifold. This theorem allows us to make analytical conclusions based on the geometry of a surface.

Initially, the homological algebra of abelian groups is studied to quantify the holes in a smooth manifold and the failure of the fundamental theorem of calculus. An isomorphism of these two cohomologies is locally established. Subsequently, sheaves and their corresponding cohomology theory are introduced to lift this local isomorphism to a global one. Ultimately, using a double diagram of sheaves, the singular and De Rham cohomologies are shown to be isomorphic, thereby proving the theorem of De Rham.