The Fourier Heat Equation in Riemannian Space.

1. Let a certain body, capable of absorbing heat, occupy a region of n dimensional Riemannian space V, whose metric is defined by the positive quadratic differential form ds2 = gjj dxl dxJ. Suppose that this body is heated by conduction in any manner. The situation can be visualized as a flow of heat from the warmer to the cooler parts of the body. The rate of flow of heat within the body can be depicted as a vector field XI(x; t) or X1(x; t), where Xi = g'JX, and Xi = gijX'. At the position x within the body for the particular instant of time t, the vector X is in the direction of the flow of heat and its magnitude is the rate of flow of heat. It is assumed that this vector field XI(x; t) or X,(x; t) is single valued and continuous with continuous partial derivatives of first order over the given region and in a given interval of time. The lines of flow of heat are the intergral solutions of the system of n ordinary differential equations of the first order