Legendre Transformation

Source

• Basically just another way to find the differential of a variable defined as a function of other variables of the same units

Examples

$U(\sigma ,V)\to H(\sigma ,P)$

• $\mathrm{d}U(\sigma ,V)=\tau \mathrm{d}\sigma -P\mathrm{d}V$
• $H(\sigma ,P)=U+PV$
• $\mathrm{d}H=\mathrm{d}U+P\mathrm{d}V+V\mathrm{d}P=\tau \mathrm{d}\sigma +V\mathrm{d}P$

Same can be done for $F(\tau ,V)$ and $G(\tau ,P)$

$\left(\mathrm{d}f(x,y)=u\mathrm{d}x+w\mathrm{d}y\right)\wedge (g\equiv f-wy)\to g(x,y)=-y^2{\left(\frac{\partial }{\partial y}\frac{f}{y}\right)}_x$

• $u={\left(\frac{\partial f}{\partial x}\right)}_y$
• $w={\left(\frac{\partial f}{\partial y}\right)}_x$