In this paper we explicitly show the equivalence between the non-Abelian
Born-Infeld action, which was proposed by Tseytlin as an effective action on
several D-branes, and its noncommutative counterpart for slowly varying fields.
This confirms the equivalence between the two descriptions of the D-branes
using an ordinary gauge theory with a constant B field background and a
noncommutative gauge theory, claimed by Seiberg and Witten. We also construct
the general forms of the 2 n-derivative terms for non-Abelian gauge fields
which are consistent with the equivalence in the approximation of neglecting (2
n+2)-derivative terms.Comment: 18 pages, LaTeX, no figures. minor corrections, version to appear in
JHE