The derivative isn't what you think it is.

Cohomology detects holes by studying when calculus rules fail on spaces. The myth “functions with zero derivative are constant” fails when holes exist. Similarly, “zero curl vector fields are gradients” fails on spaces with 1-dimensional holes like cylinders.

de Rham’s theorem establishes that the k-th homology group Hₖ(X) and k-th cohomology group H^k(X) are isomorphic. The isomorphism is given by integration: pair a k-chain c with a k-form ω by computing ∫_c ω. This pairing connects two seemingly unrelated worlds.

Differential forms unify functions, vector fields, and higher-dimensional integrands under one framework. They exist to be integrated over chains. Forms replace the separate concepts of gradient, divergence, and curl with a single exterior derivative operator.

Emmy Noether formulated the modern definition of homology using vector spaces and linear maps. Homology distinguishes different types of holes by dimension. A circle has a 1-dimensional hole while a sphere has a 2-dimensional hole. The method detects holes by testing when loops bound regions.