For a (smooth) vector bundle $F$ over a manifold $M$, one normally defines a connection to be a linear map $$ \nabla:\Gamma^{\infty}(V) \to \Omega^1(M) \otimes \Gamma^{\infty}(V), $$ satisfying $\nabla(fv) = $d$f \otimes v + f \nabla(v)$. One can then extend this to a map $$ \nabla:\Omega^k(M) \otimes_{C^{\infty}(M)} \Gamma^{\infty}(V) \to \Omega^{k+1}(M) \otimes_{C^{\infty}(M)} \Gamma^{\infty}(V) $$ by defining $$ \nabla(\omega \otimes v) = \text{d}\omega \otimes v + (-1)^k\omega \wedge \nabla(v). $$ The factor of $(-1)^k$ ensures that the definition is well-defined over the tensor product.
Alternatively, one can use the equivalent definition of a connection as a linear map $$ \nabla:\Gamma^{\infty}(V) \to \Gamma^{\infty}(V) \otimes \Omega^1(M), $$ satisfying $\nabla(vf) = v \otimes $d$f + \nabla(v)f$. However, this has the simpler extension to a map $$ \nabla:\Gamma^{\infty}(V) \otimes_{C^{\infty}(M)} \Omega^k(M) \to \Gamma^{\infty}(V) \otimes_{C^{\infty}(M)}\Omega^{k+1}(M) $$ defined by $$ \nabla(v \otimes \omega) = v \otimes \text{d}\omega + \nabla(v) \wedge \omega. $$ I.E. there is no $(-1)^k$ factor. My question is why isn't this simpler formulation the one that is normally used?
