$dS =\left(\displaystyle\frac{\partial S}{\partial U}\right)_{V,N_i} dU +\left(\displaystyle\frac{\partial S}{\partial V}\right)_{U,N_i} dV +\displaystyle\sum_i\left(\displaystyle\frac{\partial S}{\partial N_i}\right)_{U,V,N_j} dN_i$

dS = DS_UVNi * dU + DS_VUNi * dV +@SUM( DS_NiVS * dN_i , i )

$dS =\displaystyle\frac{1}{ T } dU +\displaystyle\frac{ p }{ T } dV +\displaystyle\sum_i\displaystyle\frac{ \mu_i }{ T } dN_i$

dS = dU / T + p * dV / T + @SUM( mu_i * dN_i /T , i )

$\mu_i \equiv \left(\displaystyle\frac{\partial S }{\partial N_i }\right)_{U,V,N_j}$

mu_i = DS_N_iUV

$\displaystyle\sum_i \mu_i dN_i =0$

@SUM( mu_i * dN_i , i ) = 0

$dF=\left(\displaystyle\frac{\partial F}{\partial T}\right)_{V,N_i}dT+\left(\displaystyle\frac{\partial F}{\partial V}\right)_{T,N_i}dV+\sum_i\left(\displaystyle\frac{\partial F}{\partial N_i}\right)_{T,V,N_j}dN_i$

dF = DF_TVNi * dT + DF_VTNi * dV +@SUM( DF_NiVT * N_i ), i )

$dG =\left( \displaystyle\frac{\partial G}{\partial T}\right)_{p,N_i} dT +\left(\displaystyle\frac{\partial G}{\partial p}\right)_{T,N_i} dp +\displaystyle\sum_i\left(\displaystyle\frac{\partial G}{\partial N_i}\right)_{T,p,N_j} dN_i$

dG = DG_TpNi * dT + DG_pTNi * dp + @SUM( DG_NipT * dN_i , i )

$dU =\left(\displaystyle\frac{\partial U }{\partial S }\right)_{ V , N_i } dS +\left(\displaystyle\frac{\partial U }{\partial V }\right)_{ S , N_i } dV +\displaystyle\sum_i\left(\displaystyle\frac{\partial U }{\partial N_i }\right)_{ S , V , N_j } dN_i$

dU = DU_SVNi * dS + DU_VSNi * dV + @SUM( DU_NiVS * dN_i , i )

$dU = T dS - p dV + \displaystyle\sum_i \mu_i dN_i$

dU = T * dS - p * dV + @SUM( mu_i * dN_i , i )

$dF =- S dT - p dV + \displaystyle\sum_i \mu_i dN_i$

dF =- S * dT - p * dV +@SUM( mu_i * dN_i , i )

$dG =- S dT + V dp +\displaystyle\sum_i \mu_i dN_i$

dG =- S * dT + V * dp +@SUM( mu_i * dN_i , i )