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    • 2.14: Appendix I- Integrating Factors
      https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book%3A_Thermodynamics_and_Statistical_Mechanics_(Arovas)/02%3A_Thermodynamics/2.14%3A_Appendix_I-_Integrating_Factors
      Then along each curve we have \[\begin{split} 0={dU\ns_c\over dx}&={\pz U\ns_x\over\pz x} + {\pz U\ns_c\over\pz y}\,{dy\over dx}\vph\\ &={\pz U\ns_c\over \pz x} - {A\ns_x\over A\ns_y}\,{\pz U\ns_c\ove...Then along each curve we have \[\begin{split} 0={dU\ns_c\over dx}&={\pz U\ns_x\over\pz x} + {\pz U\ns_c\over\pz y}\,{dy\over dx}\vph\\ &={\pz U\ns_c\over \pz x} - {A\ns_x\over A\ns_y}\,{\pz U\ns_c\over\pz y}\ . \end{split}\] Thus, \[{\pz U\ns_c\over\pz x}\,A\ns_y = {\pz U\ns_c\over\pz y}\,A\ns_x \equiv e^{-L} A\ns_x\,A\ns_y\ .\] This equation defines the integrating factor \(L\,\): \[L=-\ln\!\bigg({1\over A\ns_x}\,{\pz U\ns_c\over\pz x}\bigg) = -\ln\!\bigg({1\over A\ns_y}\,{\pz U\ns_c\over\pz y…
    • 2.5: Second Derivatives and Exact Differentials
      https://phys.libretexts.org/Bookshelves/Thermodynamics_and_Statistical_Mechanics/Heat_and_Thermodynamics_(Tatum)/02%3A_Partial_Derivatives/2.05%3A_Second_Derivatives_and_Exact_Differentials
      If \(z = z(x , y)\), we can go through the motions of calculating \( \frac{\partial z}{\partial x}\) and \( \frac{\partial z}{\partial y}\), and we can then further calculate the second derivatives \(...If \(z = z(x , y)\), we can go through the motions of calculating \( \frac{\partial z}{\partial x}\) and \( \frac{\partial z}{\partial y}\), and we can then further calculate the second derivatives \( \frac{\partial ^2 z}{\partial x^2}\), \(\frac{\partial ^2 x}{\partial y^2}\), \( \frac{\partial ^2 z}{\partial y \partial x}\) and \(\frac{\partial ^2 z}{\partial y \partial x}\).

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