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# hw_2 - A M,y B x,N and C M,N[Note that A M,y is...

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MS&E 303 – Fall 2004 Homework #2 (revised) – Due Friday, September 10 1. From the definition of a differential dz = ∂z ∂x y dx + ∂z ∂y x dy , and the definition of a partial differential as the limit of a finite difference, show that ∂y ∂z x = 1 / ∂z ∂y x . 2. Review Green’s Theorem from Math 192. Show that, for dz = M dx + N dy , the result ∂M ∂y x = ∂N ∂x y naturally falls out of all closed path integrals having a zero integral. 3. By any method, determine the following path integrals: (a) Z Γ 1 [2 y sin x + x cos y ] dx + [2 x sin y - y cos x ] dy (b) Z Γ 2 2 x cos y dx - x 2 sin y dy where Γ 1 is a square of edge π in the positive quadrant with one corner at the origin, and Γ 2 is the unit circle about the origin, both traversed counterclockwise. 4. The Legendre transform can be easily written when the function is explicitly known. Consider the simple function z = xy 2 . (a) Show that dz = y 2 dx + 2 xy dy . (b) What are the conjugate variables pairs [ M, x ] and [ N, y ]? (c) Write explicitly the four Legendre transforms in their natural corrdinates.
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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 = z-xM , b = z-yN and c = z-xM-yN , 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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