I am just being introduced to the notion of covectors. I see that the gradient of a scalar function is thought of as a covector. Q1) On another source I have read to think of contravariant vector field as a differential operator: Va [itex]\partial _a[/itex] and to think of a covariant vector field as differential: wadxa. From the fact that the gradient function is a covector, I thought this would be the other way around??? Q2) I'm trying to understand why df=[itex]\partial _a[/itex](f )dxa is a covector? My thoughts: [itex]\partial _a[/itex]f is a covector. dxa is a vector. So, isn't this just the definition of a contraction - multiplication of a covector and a vector to yield a scalar. Thanks very much in advance.