differential geometry - How to visualize the gradient as a one-form?

I am reading Sean Carrol's book on General Relativity, and I just finished reading the proof that the gradient is a covariant vector or a one-form, but I am having a difficult time visualizing this. I usually visualize gradients as vector fields while I visualize one-forms with level sets. How to visualize the gradient as a one-form?

Answer

If you're going to take that path, then maybe you should be thinking more of a level set density, i.e. how closely spaced the level sets in question are. Sean Carrol's book is not wonted to me: if you can get a copy of Misner Thorne and Wheeler, the first few pages do a good job of this idea with their quaint "bong" machine that sounds a "bong" bell each time a vector pierces a level set. If you can't get this readily, then the early part of Kip Thorne's lectures here is also good

Anyhow, suppose we are given a scalar field $\phi(\vec{x})$ and the tangent space $T_x\mathcal{M}$ to $x\in\mathcal{M}$ in some manifold $\mathcal{M}$, and we imagine riding along a vector $X\in T_x\mathcal{M}$ in the tangent space: how often would we pierce level sets of $\phi$: in MTW's quaint and unforgettable words (and wonderful sketches), how often would our bell sound as we rode along the vector? It would be $\nabla\phi\,\cdot\,X$ (i.e. the directional derivative). Thus $\nabla\phi$ is a dual vector to the vector space of tangent vectors. It is a linear functional $T_x\mathcal{M}\to\mathbb{R}$ on the tangent space: it takes a vector $X\in T_x\mathcal{M}$ as its input and spits out the directional derivative $\nabla\phi\,\cdot\,X$.