Vector Fields, Gradient Fields, and Potential Functions

In financial models, each point in the state space carries a "drift direction": the risk-neutral drift $\mu(S,t)$ pushes the price process in a direction that depends on the current state $(S,t)$. This is a vector field — a rule that assigns a vector to every point in a region. The most important class of vector fields in physics and mathematics are gradient fields, where the vector at each point is the gradient $\nabla\varphi$ of a scalar function $\varphi$ called the potential. Gradient fields are exactly the conservative forces (energy-conserving in physics, arbitrage-free in finance): the work done by the field depends only on the endpoints, not the path. This lesson develops vector fields, identifies gradient fields via the potential function, and sets up the conservative-field theory that underpins line integrals and Green's theorem.

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