Surface Integrals: Surface Area and Flux

The Black-Scholes PDE can be interpreted as a probability conservation law: the probability flux through the boundary of a region in the $(S,t)$-space equals the rate of probability change inside. To make this precise in three dimensions — as in a multi-factor model — you need to integrate a vector field over a two-dimensional surface embedded in $\mathbb{R}^3$. Surface integrals generalise line integrals from curves to surfaces: the scalar surface integral computes area and mass; the vector surface integral (flux) measures how much of a vector field passes through the surface per unit time. Both require the surface area element $dS=\|\mathbf{r}_u\times\mathbf{r}_v\|\,du\,dv$, which is the two-dimensional Jacobian for parameterised surfaces.

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