Editing Partial derivative (section)

===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.