## Lectures on dynamical systems by Newhouse S.

By Newhouse S.

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Additional resources for Lectures on dynamical systems

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K13693” — 2012/1/31 — 9:52 ✐ Exercise Problems ✐ 41 The thermal conductivity of insulating materials are very low. At high temperatures, the energy transfer through an insulating material may involve many modes, such as conduction through the ﬁbrous or porous solid material, conduction through the air trapped in the void spaces, and also as radiation at suﬃciently high temperatures. The heat diﬀusion equation is valid for all homogeneous media. But in heat conduction concerning composite media in which the thermophysical properties of one homogeneous region may diﬀer signiﬁcantly from those of others, the problem has to be solved by patching solutions of the heat diﬀusion equation for homogeneous medium in diﬀerent regions.

In most cases such heat sources or sinks may be speciﬁed as volumetric sources or sinks, that is, their strength may be speciﬁed per unit volume. If g˙ th represents a source strength, the total contribution from the volume element is g˙ th (δxδyδz). Therefore, for conservation of energy, [ρ (δxδyδz)] c ∂T = [ . (k ∂t T )] (δxδyδz) + g˙ th (δxδyδz) that is, ρc ∂T = ∂t . (k T ) + g˙ th This is the general form of heat diﬀusion equation. This equation, usually known as the heat equation, provides the basic tool for heat conduction analysis.

The microscopic forms of energy are those related to the molecular structure of a system and the degree of the molecular activity, and they are independent of outside reference frames. The sum of all the microscopic forms of energy is called the internal energy of a system. The energy that a system possesses as a result of its motion relative to some reference frame is called the kinetic energy. The potential energy is the energy that a system possesses as a result of its elevation in a gravitational ﬁeld.