Modern Mathematics Extends Fundamental Theorem of Calculus to Higher Dimensions

A recent social media post by Majid Hosseini highlighted that a significant portion of modern mathematics is dedicated to expanding the foundational principles of the Fundamental Theorem of Calculus (FTC) beyond a single dimension. Hosseini specifically cited Stokes' Theorem, Gauss-Bonnet theorem, De Rham Cohomology, and differential forms as key machinery in achieving this ambitious goal. The Fundamental Theorem of Calculus, a cornerstone of single-variable calculus, establishes a critical link between differentiation and integration. It essentially states that these two operations are inverse processes. This theorem allows for the calculation of definite integrals by finding antiderivatives, simplifying what would otherwise be complex summation processes. However, as mathematics progressed, the need arose to apply these concepts to more complex spaces and higher dimensions. This has led to the development of several powerful generalizations. Among these, Stokes' Theorem is a broad generalization of the FTC that relates the integral of a differential form over the boundary of an oriented manifold to the integral of its exterior derivative over the whole manifold. This theorem unifies several vector calculus theorems, including the divergence theorem and Green's theorem. Differential forms are a central concept in this generalization, providing a framework to express and manipulate functions and their derivatives in a coordinate-independent manner across various dimensions. They are essential for defining integrals on manifolds and for the formulation of theorems like Stokes' Theorem. De Rham Cohomology, another area mentioned, uses differential forms to study the topological properties of smooth manifolds, effectively linking calculus with topology. The Gauss-Bonnet theorem, while distinct, also represents a profound connection between the geometry of a surface and its topology, often expressed using differential forms and curvature. These advanced mathematical tools collectively allow mathematicians to extend the powerful insights of the FTC to intricate geometric structures and abstract spaces, revealing deeper connections within the mathematical universe.