Physical significance of one-form in a velocity field Still tentatively feeling my way through this stuff, so please go easy.
The velocity of a fluid at a point P are the components $V^{a}$ of a contravariant vector:$$v^{x},v^{y},v^{z}\equiv\frac{dx}{dt},\frac{dy}{dt},\frac{dz}{dt},$$ where $x,y,z$ are some arbitrary coordinate system. The metric tensor $g_{ab}$ for that coordinate system can be used to lower the index to give the components of a one-form:$$V_{b}=g_{ab}V^{a}.$$
This one-form will have components$$\frac{\partial\phi}{\partial x},\frac{\partial\phi}{\partial y},\frac{\partial\phi}{\partial z},$$where $\phi$ is a scalar field.
I'm assuming this one-form is just another complementary way of describing the behaviour of the fluid at P. I have some vague notion of a one-form as a set of planes or contours, each one representing different values of $\phi$. So, what is the physical significance of $\phi$? In other words, what is the relationship between $\phi$ and the velocity of the fluid?
Further thoughts – I believe the metric tensor is unitless, which means $V_{b}$ has units of $ms^{-1}$ and $\phi$ has units of $m^{2}s^{-1}$. What on Earth has units of $m^{2}s^{-1}$? 
BOUNTY EDIT – On reflection, this is easier for me to visualise in two arbitrary dimensions $(x,y)$, with a tangent vector to a parametric curve having components
$$V^{a}\equiv v^{x},v^{y}\equiv\frac{dx}{dt},\frac{dy}{dt}.$$
Lower the index with the metric to give a one-form with components$$V_{b}\equiv\frac{\partial\phi}{\partial x},\frac{\partial\phi}{\partial y}.$$What is the physical significance of the orientation, value and spacing of the constant $\phi$ contours? Thanks. 
 A: So from your definition
$$U_b = g_{ab} V^a,$$
where I have renamed the 1-form $U$ so I don't have to write coordinates to distinguish them, we see that the value of $U$ on a unit vector $W$ is the projection of the velocity $V$ onto $W$, ie. by definition
$$U_bW^b = g_{ab}V^a W^b.$$
Physically we can interpret this as saying that if we have an infinitesimal area element $\hat n dA$, where $\hat n$ is the unit normal vector and $dA$ has units of area, that the value of $U$ on $\hat n$ is the fluid flux per unit time per $dA$ through this area element.
Edits below because I mixed up my curls and divergences.
As Qmechanic mentioned, this 1-form doesn't have to be closed, unless we assume the flow is vorticity-free ie. $\nabla \times V = 0$. This is not a very physical assumption because vorticity tends to just get generated all the time, but in this case we'd be able to define a function $\phi$ such that the velocity field is a gradient: $V = \nabla \phi$.
It's more physical to assume that the flow is divergenceless, ie. that the fluid is incompressible, which could be very physical. So now the cooler thing that I wanted to mention anyway actually becomes necessary to express this condition compactly.
The way to do it is not to speak in terms of $U$ but in terms of a 2-form (or 1-form in 2d, in general it's Poincar\'e dual to $U$)
$$F_{bc} = \epsilon_{abc}V^a$$
whose value on any surface is the flux per unit time through that surface. Here $\epsilon$ is the volume form or Levi-Civita tensor. A useful conservation law for incompressible fluids is $dF = 0$, meaning that in the absence of sources or sinks, the fluid flux through a closed surface is always zero. You can think of this as a Gauss law, which makes fluid flow in 3d look a bit like a gauge theory (with gauge group $\mathbb{R}$), where the flow lines behave like magnetic field lines. Anyway, that's a story for another time. Here's some good reading though.
