The lie derivative of the scalar field $f(x)$ under $x' =x + \varepsilon v(x)$ can be calculated by noting $$f(x') = f(x + \varepsilon v) = f(x) + \varepsilon v \partial f$$ giving $$\mathcal{L}_v(f) = \varepsilon v \partial f.$$
Trying the same thing on a covector as displayed here
http://www.pas.rochester.edu/~rajeev/phy413/Grav18.pdf
I should get $$w(x')dx' = w(x)dx + \varepsilon[v\partial w dx + w \partial v dx]$$ however $$w(x')dx' = w(e + \varepsilon v)d(x + \varepsilon v ) = w(e + \varepsilon v)dx + w(e + \varepsilon v)\varepsilon dv = [w(x) + \varepsilon v \partial w ]dx + [w(x) + \varepsilon v \partial w ] \varepsilon v $$
does not reduce to this. How do I do it this way, so that I can also apply it to $g_{ab}dx^a dx^b$?
Also, is there a way to apply this same method to a contravariant vector field?