Solutions

In[51]:=

(*1*)U[S_, V_, N_] := T S - p V + μ NDt[U[S, V, N]] Dt[U[S, V, N], Constants {T, p, μ}]

Out[52]=

μ Dt[N] - V Dt[p] + T Dt[S] + S Dt[T] - p Dt[V] + N Dt[μ]

Out[53]=

μ Dt[N, Constants {p, T, μ}] + T Dt[S, Constants {p, T, μ}] - p Dt[V, Constants {p, T, μ}]

In[54]:=

(*2*)D[U[S, V, N], S] D[U[S, V, N], V]

Out[54]=

T

Out[55]=

-p

In[56]:=

(*3*)Dt[U[S, V, N], S] Dt[U[S, V, N], V] Dt[U[S, V, N], S, Constants {T, p, μ}] Dt[U[S, V, N], V, Constants {T, p, μ}]

Out[56]=

T + μ Dt[N, S] - V Dt[p, S] + S Dt[T, S] - p Dt[V, S] + N Dt[μ, S]

Out[57]=

-p + μ Dt[N, V] - V Dt[p, V] + T Dt[S, V] + S Dt[T, V] + N Dt[μ, V]

Out[58]=

T + μ Dt[N, S, Constants {p, T, μ}] - p Dt[V, S, Constants {p, T, μ}]

Out[59]=

-p + μ Dt[N, V, Constants {p, T, μ}] + T Dt[S, V, Constants {p, T, μ}]

In[60]:=

(*4*)Dt[U[S, V, N]]/.{Dt[T] 0, Dt[p] 0, Dt[μ] 0}

Out[60]=

μ Dt[N] + T Dt[S] - p Dt[V]

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