In this chapter, we will discuss a variety of physical problems that can be modeled by integral equations, typically over planar regions or spatial regions. We will see that Chapter 7’s divergence theorem, Stokes’ theorem, and change of variable formulas enable us to turn integral equations into partial differential equations (PDEs). In Chapters 11, 16, and 17, we will learn methods for getting solution formulas for PDEs,

and in Chapters 12 and 14, wewill learnmethods for numerical approximation of solutions of PDEs. Our first physical applications are to flow of mass or heat energy. Recall from Example 7.36 in Section 7.5 that

v • n̂ dS

is the rate of mass flow out of S. Here is the mass density, in units of mass per volume, v is the velocity vector field of the fluid; and S is a constant, that is, time-independent, closed, piecewise smooth, oriented surface enclosing a solid V having outward unit normal vector n̂. We may refer briefly to such a situation as S encloses a “control” volume V . In a physical problem in which a substance can flow through the boundary of a control

volume, the divergence theorem can relate that flow, which is an integral over the surface bounding the control volume, to an integral over the control volume. Define M = V dV to be the total mass of the fluid in the solid V . In Example 7.39 in

Section 7.6, we saw that

v • n̂ dS = −dM dt

= − V

∂t dV.