The Fundamental Theorem in Higher Dimensions: Unification and Finance Bridge

Every theorem in this unit — the FTC, Green's theorem, Stokes's theorem, and the Divergence theorem — says the same thing in a different number of dimensions: the integral of a "derivative" over a region equals the integral of the original quantity over the region's boundary. In the language of differential forms, this is a single identity $\int_{\partial\Omega}\omega=\int_\Omega d\omega$, called the Generalised Stokes's Theorem. Recognising this structure matters for quant finance because it is the mathematical backbone of everything from the Feynman-Kac formula to the PDE derivation of Black-Scholes: every time we convert a boundary condition on an option's payoff into a partial differential equation governing its interior price, we are applying the generalised FTC in an infinite-dimensional setting.

Learning objectives. By the end of this note you should be able to: