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344 Advanced~Calculus, Fifth Editionwhere p is the density. Equating the two expressions, one finds10 (cp - k div grad T (5.143)RSince this must hold for an arbitrary solid region R, the function integrated(if continuous) must be zero everywhere. HenceaTcp- - k div grad T = 0.at(5.144)This is the fundamental equation for heat conduction. If the body is intemperature equilibrium, aT/at = 0, and one concludes thatdiv grad T = 0; (5.145)that is, T is hamtonic.e) Thermodynamics. Let a certain volume V of a gas be given, enclosed in acontainer, subject to a pressure p. It is known from experiment that for eachkind of gas there is an "equation of state"connecting pressure, volume, and temperature T. For an "ideal gas" (lowdensity and high temperature), Eq. (5.146) takes the special form:pV = RT, (5.147).*where R is constant (the same for all gases, if one mole of gas is used).With each gas is also associated a scalar U, the total internal energy; thisis analogous to the kinetic energy plus potential energy considered above.For each gas U is given as a definite function of the "state," hence of p andv:U = U(p, V). (5.148)The particular equation (5.148) depends on the gas considered. For an idealgas, (5.148) takes the form:where cv is constant, the specific heat at constant volume.A particularprocess gone through by a gas is a succession of changes inthe state, so that p and V, and hence T and U, become functions of time t.The state at time t can be represented by a point (p, V) on a pV diagram(Fig. 5.38) and the process by a curve C, with t as parameter. During such aprocess it is possible to measure the amount of heat, Q, received by the gas.
Chapter 5 Vector Integral Calculus 345Figure 5.38Thermodynamic processes.The first law of thermodynamics is equivalent to the statement thatwhere Q(t) is the amount of heat received up to time t.Hence the amount of heat introduced in a particular process is given byan integral:Now by (5.148), dU is expressible in terms of dp and dV:Hence (5.151) can be written as a line integral:or, with (5.152) understood, simply thus: . e .For (5.153) or (5.154) to be independent of the path C, one must havea auav ap = -(- +PI;;iapavthat is,"("1a2ua2u-- -- + avap apav 1." '
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3.6 Combined Operations 183*3.7 Cur
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Case of Two Particles 662Case of N
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Differential Calculusof Functionsof
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Vector Integral CalculusPART I.TWO-
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