Gradient is covector or not? according to Chriseigen, we need to differentiate Delf(gradient f) and df(differential f).

To convert vector to covector, metric tensor is needed as shown above. It’s easy to understand the vector dot equals vector component time metric tensor, however, seems the epsilon j is added from no where. The explanation is that it is in the dual space, so the basis is epsilon hence needed. (? will be clarified later on)

Next Eigenchris went deeper to prove the above conclusion again.