ch7 - Chapter Seven Harmonic Functions 7.1 The Laplace...

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Chapter Seven Harmonic Functions 7.1. The Laplace equation. The Fourier law of heat conduction says that the rate at which heat passes across a surface S is proportional to the flux, or surface integral, of the temperature gradient on the surface: k  S T dA . Here k is the constant of proportionality, generally called the thermal conductivity of the substance (We assume uniform stuff. ). We further assume no heat sources or sinks, and we assume steady-state conditions—the temperature does not depend on time. Now if we take S to be an arbitrary closed surface, then this rate of flow must be 0: k  S T dA 0. Otherwise there would be more heat entering the region B bounded by S than is coming out, or vice-versa. Now, apply the celebrated Divergence Theorem to conclude that  B T dV 0, where B is the region bounded by the closed surface S . But since the region B is completely arbitrary, this means that T 2 T x 2 2 T y 2 2 T z 2 0. This is the world-famous Laplace Equation . Now consider a slab of heat conducting material, 7.1
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in which we assume there is no heat flow in the z -direction. Equivalently, we could assume we are looking at the cross-section of a long rod in which there is no longitudinal heat flow. In other words, we are looking at a two-dimensional problem—the temperature depends only on x and y , and satisfies the two-dimensional version of the Laplace equation: 2 T x 2 2 T y 2 0. Suppose now, for instance, the temperature is specified on the boundary of our region D , and we wish to find the temperature T x , y in region. We are simply looking for a solution of the Laplace equation that satisfies the specified boundary condition. Let’s look at another physical problem which leads to Laplace’s equation. Gauss’s Law of electrostatics tells us that the integral over a closed surface S of the electric field E is proportional to the charge included in the region B enclosed by S . Thus in the absence of any charge, we have  S E dA 0.
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ch7 - Chapter Seven Harmonic Functions 7.1 The Laplace...

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