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**Unformatted text preview: **p is a function of volume V 1 2 ± and internal energy U , p = h ( V, T ) = h ( V, t ( V, U )) = g ( V, U ). So to fnd the partial derivative oF p with respect to V , we have to specify the Function we = ∂h are reFerring to: h or g . We write ( ∂p = ∂g and ( ∂p . Below we ∂V ) U ∂V ∂V ) T ∂V will see how such partial derivatives are related. By ( ∂f ∂x ) y we mean the partial derivative oF f with respect to x when f is seen as a Function oF variables x and y . Theorem 4 (Chain Rule) Let w, u, v, x, y be physical variables such that there exist diﬀerentiable functions f, g, h such that w = f ( u, v ) , u = g ( x, y ) and v = h ( x, y ) . Then ± ∂w = ∂f ∂g + ∂f ∂h ∂x y ∂u ∂x ∂v ∂x ± ∂w = ∂f ∂g + ∂f ∂h ∂y x ∂u ∂y ∂v ∂y IF we take w = f ( u, v ), u = g ( x, y ), and v = h ( x ), then applying the Chain Rule we get ∂w ∂f ∂g = , ∂y x ∂u ∂y ∂h since = 0. ∂y 2...

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