I have trouble of understanding Lee's textbook, Introduction to smooth manifold, about Lie derivatives for differential forms.
First let me state the content in textbook.
Let $A$ be a smooth contravariant $k$-tensor field on a smooth manifold $M$, and let $V, X_1, \dots, X_k$ be smooth vector fields on $M$. Then
$$\mathcal{L}_V(A(X_1, \dots, X_k)) = (\mathcal{L}_VA)(X_1, \dots, X_k) + \sum_{i=1}^kA(X_1, \dots, X_{i-1}, \mathcal{L}_VX_i, X_{i+1}, \dots, X_k).$$
Then what i want to prove is starting from this obtain
$$(\mathcal{L}_VA)(X_1, \dots, X_k) = V(A(X_1, \dots, X_k)) - \sum_{i=1}^kA(X_1, \dots, X_{i-1}, [V, X_i], X_{i+1}, \dots, X_k).$$
Textbook states it is nothing but application of
$\mathcal{L}_Vf = Vf$ for any smooth function $f$ on $M$ and $\mathcal{L}_VX = [V, X]$ for any vector field $X$ on $M$.
Now i got confusion, apparently, i can write
\begin{align} (\mathcal{L}_VA)(X_1, \dots, X_k) &= (\mathcal{L}_VA)(X_1, \dots, X_k) + \sum_{i=1}^kA(X_1, \dots, X_{i-1}, \mathcal{L}_VX_i, X_{i+1}, \dots, X_k) \\ & = V(A(X_1, \dots, X_k)) + \sum_{i=1}^kA(X_1, \dots, X_{i-1}, [V, X_i], X_{i+1}, \dots, X_k) \end{align} Here i have a trouble with the $+$ sign for the second term. The textbook and other references $i.e$, wiki
How $-$ sign comes from?
I obtain the same formula from explicit computation from the Lie derivatives (via pullback, $\mathcal{L}_V w = \lim_{t\rightarrow 0} \frac{\phi_t^* w -w}{t}$) but i want to follow the textbook's approach.