298 Chapter 5 Higher Dimensions and Other Coordinates
2.Suppose that the frame is circular and that its equation isx^2 +y^2 =a^2.
Write an initial value–boundary value problem for a membrane on a circu-
lar frame. (Use polar coordinates.)
3.What should the three-dimensional wave equation be?5.2 Three-Dimensional Heat Equation
Vector Derivation
To illustrate a different technique, we are going to derive the three-dimensional
heat equation using vector methods. Suppose we are investigating the temper-
ature in a body that occupies a regionRin space. (See Fig. 5.) LetVbe a
subregion ofRbounded by the surfaceS. The law of conservation of energy,
applied toV,says
net rate of heat in+rate of generation inside=rate of accumulation.Our next job is to quantify this statement. The heat flow rate at any point
insideRis a vector function,q,measuredinJ/m^2 s or similar units. The rate of
heat flow through a small piece of the surfaceSwith area Ais approximately
nˆ·q A(see Fig. 6), wherenˆis the outward unit normal. This quantity is
positive for outward flow, so the inflow is its negative. The net inflow over the
entire surfaceSis a sum of quantities like this, which becomes, in the limit as
Ashrinks, the integral
∫∫
S−q·nˆdA.The term “rate of generation inside” in the energy balance is intended to in-
clude conversion of energy from other forms (chemical, electrical, nuclear) to
thermal. We assume that it is specified as an intensitygmeasured in J/m^3 sor
Figure 5 A solid body occupying a regionRin space and a subregionVwith
boundaryS.