A differential of the form
(1)
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is exact (also called a total differential) if is path-independent. This will be true if
(2)
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so and must be of the form
(3)
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But
(4)
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(5)
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so
(6)
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There is a special notation encountered especially often in statistical thermodynamics. Consider an exact differential
(7)
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Then the notation , sometimes referred to as constrained variable notation, means "the partial derivative of with respect to with held constant." Extending this notation a bit leads to the identity
(8)
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where it is understood that on the left-hand side is treated as a variable that can itself be held constant.