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Partial derivative
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===Thermodynamics, quantum mechanics and mathematical physics=== Partial derivatives appear in thermodynamic equations like [[Gibbs-Duhem equation]], in quantum mechanics as in [[Schrödinger equation|Schrödinger wave equation]], as well as in other equations from [[mathematical physics]]. The variables being held constant in partial derivatives here can be ratios of simple variables like [[mole fraction]]s {{math|''x<sub>i</sub>''}} in the following example involving the Gibbs energies in a ternary mixture system: <math display="block">\bar{G_2}= G + (1-x_2) \left(\frac{{\partial G}}{{\partial x_2}}\right)_{\frac{x_1}{x_3}} </math> Express [[mole fraction]]s of a component as functions of other components' mole fraction and binary mole ratios: <math display="inline">\begin{align} x_1 &= \frac{1-x_2}{1+\frac{x_3}{x_1}} \\ x_3 &= \frac{1-x_2}{1+\frac{x_1}{x_3}} \end{align}</math> Differential quotients can be formed at constant ratios like those above: <math display="block">\begin{align} \left(\frac{\partial x_1}{\partial x_2}\right)_{\frac{x_1}{x_3}} &= - \frac{x_1}{1-x_2} \\ \left(\frac{\partial x_3}{\partial x_2}\right)_{\frac{x_1}{x_3}} &= - \frac{x_3}{1-x_2} \end{align}</math> Ratios X, Y, Z of mole fractions can be written for ternary and multicomponent systems: <math display="block">\begin{align} X &= \frac{x_3}{x_1 + x_3} \\ Y &= \frac{x_3}{x_2 + x_3} \\ Z &= \frac{x_2}{x_1 + x_2} \end{align}</math> which can be used for solving [[partial differential equation]]s like: <math display="block">\left(\frac{\partial \mu_2}{\partial n_1}\right)_{n_2, n_3} = \left(\frac{\partial \mu_1}{\partial n_2}\right)_{n_1, n_3}</math> This equality can be rearranged to have differential quotient of mole fractions on one side.
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