Cheat sheets/Thermodynamics in differential form

Internal energy

$U = U (T, V)$
$d U = {(\frac{\partial U}{\partial T})}_{V} d T + {(\frac{\partial U}{\partial V})}_{T} d V$
$d U = d Q + d W$
$d U = C_{V} d T + {(\frac{\partial U}{\partial V})}_{T} d V$

$H = H (T, P)$
$d H = {(\frac{\partial H}{\partial T})}_{P} d T + {(\frac{\partial H}{\partial P})}_{T} d P$
$d H = C_{P} d T + {(\frac{\partial H}{\partial P})}_{T} d P$

${(\frac{\partial T}{\partial V})}_{S} = - {(\frac{\partial P}{\partial S})}_{V}$
${(\frac{\partial T}{\partial P})}_{S} = {(\frac{\partial V}{\partial S})}_{P}$
${(\frac{\partial S}{\partial V})}_{T} = {(\frac{\partial P}{\partial T})}_{V}$
${(\frac{\partial S}{\partial P})}_{T} = - {(\frac{\partial V}{\partial T})}_{P}$