In mathematics, an "'exact differential equation "'or "'total differential equation "'is a certain kind of ordinary differential equation which is widely used in physics and engineering.
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Following the definition above in formula ( 1 ), for such a fictive'reversible'process, a quantity of transferred heat ( an inexact differential ) is analyzed as a quantity, with ( an exact differential ):
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For example, thermodynamic work is a holonomic process function since the integrating factor \ lambda = 1 / p ( where p is pressure ) will yield exact differential of the volume state function dV = \ delta W / p.
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Where is the infinitesimal heat flow into the system, and is the infinitesimal work done by the system, is the chemical potential of particle type and is the number of type particles . ( Note that neither nor are exact differentials.
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It is commonly used to solve ordinary differential equations, but is also used within multivariable calculus when multiplying through by an integrating factor allows an inexact differential to be made into an exact differential ( which can then be integrated to give a scalar field ).
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As recounted below, in the section headed Entropy, the second law of thermodynamics observes that if heat is supplied to a system in which no irreversible processes take place and which has a well-defined temperature, the increment of heat and the temperature form the exact differential
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In the above, the letter " d " indicates an exact differential, expressing that internal energy " U " is a property of the state of the system; they depend only on the original state and the final state, and not upon the path taken.
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The second law of thermodynamics as stated by Carath�odory essentially amounts to the statement that heat is a holonomic process function since the integrating factor \ lambda = 1 / T ( where T is temperature ) will yield the exact differential of an entropy state function dS = \ delta Q / T.
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It can be seen that the integrand can be expressed as the exact differential of the function P ( t ) V ( t ) and that therefore, the integral can be expressed as the difference in the value of P ( t ) V ( t ) at the end points of the integration.
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Forms which are the image of other forms under the exterior derivative, plus the constant function in are called "'exact "'and forms whose exterior derivative is are called "'closed "'( see closed and exact differential forms ); the relationship 0 } } then says that exact forms are closed.
