Unformatted text preview: A ( M,y ), B ( x,N ) and C ( M,N ). [Note that A ( M,y ) is pathological and will have an extremely simple answer. This is a result of using such a simple and easily tractable function for z ( x,y ).] (d) Write the diﬀerentials corresponding to dA , dB and dC . (e) Explain and show that ± ∂B ∂N ¶ x = ± ∂C ∂N ¶ M . List all the equivalent relationships that should exist between derivatives of z,A,B and C . 5. Given a function z, and its Legendre transforms a = zxM , b = zyN and c = zxMyN , show why following relationships hold and what they both equal (these are simple results from the deﬁnition of the diﬀerentials): (a) ± ∂z ∂x ¶ y = ± ∂b ∂x ¶ N (b) ± ∂a ∂M ¶ y = ± ∂c ∂M ¶ N (c) ± ∂M ∂y ¶ x = ± ∂N ∂x ¶ y (d) ± ∂z ∂y ¶ x = ± ∂a ∂y ¶ M (e) ± ∂b ∂N ¶ x = ± ∂c ∂N ¶ M (f) ± ∂x ∂N ¶ M = ± ∂y ∂M ¶ N 1...
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 Fall '04
 THOMPSON
 Calculus, Derivative, path integrals, following path integrals

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