  For a polymatroid $P$ over $[n]$, Bernardi, K\'{a}lm\'{a}n and Postnikov
[\emph{Adv. Math.} 402 (2022) 108355] introduced the polymatroid Tutte
polynomial $\mathscr{T}_{P}$ relying on the order $1<2<\cdots<n$ of $[n]$,
which generalizes the classical Tutte polynomial from matroids to polymatroids.
They proved the independence of this order by the fact that $\mathscr{T}_{P}$
is equivalent to another polynomial that only depends on $P$. In this paper,
similar to the Tutte's original proof of the well-definedness of the Tutte
polynomial defined by the summation over all spanning trees using activities
depending on the order of edges, we give a direct and elementary proof of the
well-definedness of the polymatroid Tutte polynomial.
