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The gauge field strength tensor $F_{\mu \nu}$ in Abelian and non-Abelianlocal gauge field theories is a key object in the construction of theLagrangian since it provides the kinetic term(s) of the gauge field(s) $A_\mu$.When introducing this object, most of textbooks employ as a tool the commutatorof the gauge covariant derivatives $D_\mu \psi$ of a fermion field $\psi$:$F_{\mu \nu} \psi = (i/g)[D_\mu,D_\nu]\psi$. I argue that such a construction,although completely correct and valid, is not pedagogical enough for manystudents that approach the gauge field theories for the first time. Anotherconstruction, based on the object $D_\mu A_\nu$: $F_{\mu \nu} = D_{[\mu}A_{\nu]}$, which avoids the introduction of additional and, for the case inconsideration, spurious degrees of freedom such as the fermion one, simpler,more pedagogical in many cases, and suitable for first-time students, ispresented.

A New Pedagogical Way of Finding Out the Gauge Field Strength Tensor in Abelian and Non-Abelian Local Gauge Field Theories