Please help. I do understand the representation of a vector as: vi∂xi I also understand the representation of a vector as: vidxi So far, so good. I do understand that when the basis transforms covariantly, the coordinates transform contravariantly, and v.v., etc. Then, I study this thing called the gradient. If I work out an indicial notation, I get this: ∇f=(∂f/∂xi)dxi Now comes my trouble. I can "envision" in my mind, that ∂xi are tangent to the coordinate curves and are, essentially, directions. But I cannot see the "directions" of dxi I cannot see them as "basis vectors" as easily as I see ∂xi I do understand that dxi ∂xj = δij And I understand how the direction of, say in 3D space, dx1 is perpedicular to the plane formed by ∂x2 and ∂x3. But I cannot easily see the "directions" of the basis of the dual vectors as easily as I can see the basis of the original vectors (as tangent to the coordinate curves). I cannot make the leap and replace dxi by e1, e2, e3, and easily as I can replace ∂xi with e1, e2, e3 I can begin with the definition of how I construct a dual basis... But I cannot easily make the leap to see the basis as this form dxi. I just don't see "directions" here. Can someone provide some insight? Also, given a metric, I can convert the basis of the gradient to a covariant form and the components to a contravariant. So why is the gradient called contravariant, when it can go either way with a metric?