Introduction

In the theory of heat conduction the assumption is made that heat flows in the direction of decreasing temperature. We also assume that the time rate at which heat flows across a surface area is proportional to the component of the temperature gradient in the direction perpendicular to the surface area. If the temperature does not depend on time, then is given by the vector

where K is the thermal conductivity of the medium and it is assumed to be constant.

If the domain in which T(x,y) is defined is simply connected, then a conjugate harmonic function S(x,y) exists, and

is an analytic function. The curves are called isothermals and are lines connecting points of the same temperature. The curves are called the heat flow lines, and one can visualize the heat flowing along these curves from points of higher temperature to pints of lower temperature.

Boundary value problems for steady state temperatures are realizations of the Dirichelt problem where the value of the harmonic function T(x,y) is interpreted as the temperature at the point (x,y).

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(c) John Mathews, 1998, 2006