Dual vector bundle in differential geometry I am a graduate student in control theory. The entity "vector bundle" appears oftentimes. For instance, the tangent vector bundle $TM$ is the set of local tangent vector spaces at coordinate $x$ of a certain manifold $M$. It utilizes the "vector bundle" $\xi$ with total manifold $E_\xi$ (belonging to which I do not grasp) in the case of a fiber $\xi_p$, for coordinate $p \in M$ and a section $\Gamma(\xi)$, as the name recalls, the applications around the neighborhood of certain coordinate $p$ belonging to fiber $\xi_p$.
I do not comprehend, however, what a dual vector bundle means. The article available on here on page 140, says the following, in german:
Let application $X \in \Gamma(TM)$ and $s \in \Gamma_p(\xi)$. Then, for each vector $v \in T_p M$, there is an element $\nabla_v s \in \xi$ unequivocally defined by equality $\nabla_v s := (\nabla_X s)_p$, for $X_p = v$. Let $\xi^*$ be a $\xi$ dual vector bundle, so there is for each $\omega \in \Gamma_p(\xi^*)$ and each $v \in T_p M$ exactly one element $\nabla_v \omega \in \xi_p^*$ with:
$v \cdot \langle \omega, s \rangle = \langle \nabla_v \omega, s \rangle + \langle \omega, \nabla_v s \rangle$.
 A: The total manifold of the bundle is just the underlying manifold to the vector bundle, forgetting the projection and the base.
If you assume for a second that your vector bundle $\xi$ as a sub-bundle of a trivial bundle $U\times M$ the total manifold of $\xi$ is just
$$\{ (v,p) \in U \times M \mid v \in \xi_p \}$$
The dual bundle to $\xi$ is the vector bundle deduced from $\xi$ by replacing each fibre $\xi_p$ by its dual $\xi'_p$, the vector space of linear forms on $\xi_p$. Other canonical constructions made “fibre by fibre” yield more fibre bundles: taking cartesian products, symmetric or exterior powers, or looking for the zero hyperplan of a linear form, for instance.
A difficulty to begin in this topic is that the “synthetic” approach used in the article summons a great many different notions. All of them are really needed, and in the end, it is much easier than the “coordinates” approach, since it allows to reason geometrically rather than to reduce everything to a coordinate computation.  If that helps, it is useful to translate “in coordinates” for a while — and that is most of the time mandatory to work out examples.
Suggested reads: Michael Spivak, Jacques Lafontaine (Introduction to differential Manifolds), but really speak with your professor before spending a long time in the books, so that they can help you to make an appropriate use of them, especially regarding the balance between understanding the vocabulary of differential geometry and bee-lining to the topic you are actually interested in.
A: In the standard basis in $\mathbb{R}^n$, we have $\partial_i=\frac{\partial}{\partial x^i}$ (ith directional derivative/ coordinate vector), dually we have $\partial^i = dx^i$ (ith component function/ increment). They interact via natural pairing:
$$\partial^i(\partial_j) = \delta^i_j = \bigg\{\begin{matrix}1&&\text{ if i=j}\\0&&\text{otherwise}\end{matrix}.$$

For example:
Let $v = \partial_1+2\partial_2$ and suppose $\omega = -\partial^1+4\partial^2.$
Then:
$$\omega(v) = (-\partial^1+4\partial^2)(\partial_1+2\partial_2) = -\partial^1\partial_1 -2\partial^1\partial_2 +4\partial^2\partial_1+8\partial^2\partial_2 = -1 - 0 + 0 + 8 = 7.$$

Now, suppose $X\in \Gamma(TM)$ (i.e. $X$ is a vector field) and $\omega\in \Gamma(T^*M)$ (i.e. $\omega$ is a dual vector field), then at each point, $p\in M$ we have a chart $\varphi:U_p\to \mathbb{R}^n$ such that the pullback of $\omega_p$ through the inverse and the pushforward of $X_p$ interact as in the above example (down in $\mathbb{R}^n$):
$$(\varphi^{-1})^*(\omega_p)\bigg(\varphi_*(X_p)\bigg).$$
Alternatively, if we induce coordinate frames on the bundles (or otherwise just have them from context), we may write out the interaction over the manifold itself: $\omega_p(X_p)$ and evaluate according to the kronecker-delta expression ($\delta^i_j$) between the frames.

This was in attempt to illustrate how duality works in basic differential geometry (involving tangent and cotangent vector bundles). The idea should be roughly the same for other vector bundles and their duals. I hope this edit helps!
