CPRT/b380

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Energie I (eine neue Variable)

$N$ kommt, $\mu$ geht.

$(01)$	$dE(N,S,V)$	$=$	$+\mu dN+tdS+pd(-V)$
$(02)$	$dI(\mu ,S,V)$	$=$	$-Nd\mu +tdS+pd(-V)$
$(02)$	$dI(N,S,V)$	$=$	$+{\mu }_E^{I}dN+t_E^{I}dS+p_E^{I}d(-V)$
$(03)$	$I(\mu ,S,V)$	$=$	$+tS+p(-V)$
$(03)$	$I(N,S,V)$	$=$	$+{\mu }_E^{I}N+t_E^{I}S+p_E^I(-V)$

Variante a)

(04a) folgt aus (03) und (03a). (05a) folgt aus (02a) und (03a).

$(01a)$	$-Nd\mu (N,S,V)$	$=$	$+[0-N\frac{\partial \mu }{\partial N}]dN+[0-N\frac{\partial \mu }{\partial S}]dS+[0-N\frac{\partial \mu }{\partial (-V)}]d(-V)$
$(02a)$	$dI(N,S,V)$	$=$	$+[0-N\frac{\partial \mu }{\partial N}]dN+[t-N\frac{\partial \mu }{\partial S}]dS+[p-N\frac{\partial \mu }{\partial (-V)}]d(-V)$
$(03a)$	$F(N,S,V)$	$=$	$+[0-N\frac{\partial \mu }{\partial N}]N+[t-N\frac{\partial \mu }{\partial S}]S+[p-N\frac{\partial \mu }{\partial (-V)}](-V)$
$(04a)$	$0(N,S,V)$	$=$	$+[0-N\frac{\partial \mu }{\partial N}]N+[0-N\frac{\partial \mu }{\partial S}]S+[0-N\frac{\partial \mu }{\partial (-V)}](-V)$
$(05a)$	$0(N,S,V)$	$=$	$-Nd[0-N\frac{\partial \mu }{\partial N}]-Sd[t-N\frac{\partial \mu }{\partial S}]-(-V)d[p-N\frac{\partial \mu }{\partial (-V)}]$

Variante b) (am einfachsten zu merken)

(04b) folgt aus (03) und (03b). (05b) folgt aus (02b) und (03b).

$(01b)$	$-Nd\mu (N,S,V)$	$=$	$+[0-N\frac{\partial \mu }{\partial N}]dN+[0-N\frac{\partial t}{\partial N}]dS+[0-N\frac{\partial p}{\partial N}]d(-V)$
$(02b)$	$dF(N,S,V)$	$=$	$+[0-N\frac{\partial \mu }{\partial N}]dN+[t-N\frac{\partial t}{\partial N}]dS+[p-N\frac{\partial p}{\partial N}]d(-V)$
$(03b)$	$F(N,S,V)$	$=$	$+[0-N\frac{\partial \mu }{\partial N}]N+[t-N\frac{\partial t}{\partial N}]S+[p-N\frac{\partial p}{\partial N}](-V)$
$(04b)$	$0(N,S,V)$	$=$	$+[0-N\frac{\partial \mu }{\partial N}]N+[0-N\frac{\partial t}{\partial N}]S+[0-N\frac{\partial p}{\partial N}](-V)$
$(05b)$	$0(N,S,V)$	$=$	$-Nd[0-N\frac{\partial \mu }{\partial N}]-Sd[t-N\frac{\partial t}{\partial N}]-(-V)d[p-N\frac{\partial p}{\partial N}]$

Variante c)

(04c) folgt aus (03) und (03c). (05c) folgt aus (02c) und (03c).

$(01c)$	$-Nd\mu (N,S,V)$	$=$	$+[0+S\frac{\partial \mu }{\partial S}+V\frac{\partial \mu }{\partial V}]dN+[0+S\frac{\partial t}{\partial S}+V\frac{\partial t}{\partial V}]dS+[0+S\frac{\partial p}{\partial S}+V\frac{\partial p}{\partial V}]d(-V)$
$(02c)$	$dF(N,S,V)$	$=$	$+[0+S\frac{\partial \mu }{\partial S}+V\frac{\partial \mu }{\partial V}]dN+[t+S\frac{\partial t}{\partial S}+V\frac{\partial t}{\partial V}]dS+[p+S\frac{\partial p}{\partial S}+V\frac{\partial p}{\partial V}]d(-V)$
$(03c)$	$F(N,S,V)$	$=$	$+[0+S\frac{\partial \mu }{\partial S}+V\frac{\partial \mu }{\partial V}]N+[t+S\frac{\partial t}{\partial S}+V\frac{\partial t}{\partial V}]S+[p+S\frac{\partial p}{\partial S}+V\frac{\partial p}{\partial V}](-V)$
$(04c)$	$0(N,S,V)$	$=$	$+[0+S\frac{\partial \mu }{\partial S}+V\frac{\partial \mu }{\partial V}]N+[0+S\frac{\partial t}{\partial S}+V\frac{\partial t}{\partial V}]S+[0+S\frac{\partial p}{\partial S}+V\frac{\partial p}{\partial V}](-V)$
$(05c)$	$0(N,S,V)$	$=$	$-Nd[0+S\frac{\partial \mu }{\partial S}+V\frac{\partial \mu }{\partial V}]-Sd[t+S\frac{\partial t}{\partial S}+V\frac{\partial t}{\partial V}]-(-V)d[p+S\frac{\partial p}{\partial S}+V\frac{\partial p}{\partial V}]$