Should it not be $\Big(F'(x)v \Big)^j = \frac{\partial F^j}{\partial x^k }(x)v^k$?
Then also how is $F^*dy^j = \frac{\partial F^j}{\partial x^i}dx^i$ derived? I cannot what $\beta_j$ has been taken to be. Surely $\beta_j$ would be equal to $dy^j$ but this does not seem to work. Are we assuming that $F(x)=y$?
Correct, it should be $\left(F'(x)v\right)^j = \frac{\partial F^j}{\partial x^k}v^k$
To get $F^* dy^j = \frac{\partial F^j}{\partial x^i} dx^i$, just let $\beta_j$ be the constant function $1$