Musical isomorphisms and 1-forms In this lecture by James Cook musical morphisms are introduced:
The exterior differential of a function (function corresponding to a 0-form) results in a 1-form (otherwise referred to as a work form).
The gradient at every point of a manifold would form a field
$$\vec F =\sum F^i \frac{\partial}{\partial x^i}\bigg\vert_p$$
which would yield the work form $\omega$ at a given point through the operation
$$\omega_{\vec F}=\flat \vec F$$
resulting in
$$\omega = \omega_1 dx + \omega_2 dy + \omega_3 dz.$$
[I modified the notation from the original recording from $\alpha$ to $\omega$, although this may wrong.]

Is the flat $\flat$ simply making reference to the covectors, $dx_i$ or $\mathrm e^i$ elements in the cotangent bundle?

 A: The symbol $\flat$ is used to denote the isomorphism between vectors and covectors provided by a metric on a given vector space $g: V \times V \rightarrow \mathbb{R}$ in particular,
$$ \flat v = g(v, \cdot) \qquad \text{which means} \qquad \flat v = \alpha \ \ \& \ \ \ \ \alpha (x) = g(v,x) $$
for each $x \in V$. This construction naturally extends to a manifold with a metric tensor field since the metric tensor provides a metric for the tangent space at a given point in the manifold. Notice, this could be a Riemannian manifold if the metric is positive definite, but generally we only insist the metric be nondegenerate and symmetric. Sylvester's Law of Inertia tells us that there exists a coordinate system in which the metric has a formula of the form:
$$ g(x,y) = x^1y^1+ \cdots + x^ry^r-x^{r+1}y^{r+1}- \cdots -x^ny^n = \sum x^iy_i (\text{using notation below defined})$$
for example, Minkowksi metric is often given with $r=3$ and $n=4$ or sometimes we see the pattern $-+++$. Sometimes we extend the meaning of $\flat$ to the components of $v = \sum v^i \partial_i$ in the sense $\flat v^i = v_i$ where:
$$ v_i = \sum_j g_{ij}v^j $$
in other words, the $\flat$ is used to lower the indices from contravariant to covariant. This is the origin of the joke of using $\flat$ for this map and conversely $\sharp$ for $\sharp v_i = v^i$ where the inverse metric (as a matrix) $g^{ij}$ is used; $\sharp v_i = v^i =\sum_j g^{ij} v_j$. There are many things about this written in Gravitation by Misner Thorn and Wheeler, I think this is also nicely discussed in John Lee's Introduction to Smooth Manifolds, page 341-343.
