I am studying analysis on manifolds and am trying some problems and I am conceptually stuck on one. It is from Do Carmo's book Differential Forms and Applications, chapter one problem 11 b.
I'll detail where I am stuck and then put the problem into context afterwards for those interested. Consider a differentiable vector field $v: \mathbb R^n \to \mathbb R^n$, given as $v = \sum\limits_{i=1}^n a_i(x_1,\ldots,x_n)e_i $. Then I need to find the differential 1-form obtained from $v$ by the canonical isomorphism induced by the inner product $\left<\ \ ,\ \right>$. Now it is not clear to me what is meant here. Intuitively it would make most sense to me that he means that $\omega$ is the 1-form that assigns to each point $(\bar{x}_1,\ldots,\bar{x}_n)$ the linear function $$\omega(x_1,\ldots,x_n) = a_1(\bar{x}_1,\ldots,\bar{x}_n)x_1 +\ldots +a_n(\bar{x}_1,\ldots,\bar{x}_n)x_n. $$ But I am not sure how this would be induced by the inner product. Maybe someone could clear it up. As a side-note, this was not discussed earlier in the book. Now I will give some context to the problem.
We define \begin{equation} {\rm div}\ (v) = \operatorname{trace} (dv_p). \end{equation} Let $v = \sum\limits_{i=1}^n a_ie_i $. Then $ dv_p = \frac{d a_i}{dx_j}$ so that we immediately see that \begin{equation} {\rm div}\ (v) = \operatorname{trace}(dv_p) = \sum\limits_{i=1}^n \frac{a_i}{dx_i}. \end{equation} Now I need to show that the divergence can be obtained via $$ v \rightarrow \omega \rightarrow *\omega \rightarrow d(*\omega) = \operatorname{div} (v)\nu. $$ where $*$ is hodge-star operation that was defined in a previous question, and $\nu$ is the volume element of $\mathbb R^n$ defined by $\nu =dx_1 \wedge \ldots \wedge dx_n$.
If you have a vector space with a scalar product $\langle\,,\rangle$, the scalar product induces an isomorphism $$\lambda: V\rightarrow V^*$$ by letting $$\lambda(v)(w) := \langle v, w\rangle$$
While $V$ and $V^*$ are always isomorphic, there is no natural choice of an isomorphism in general. If you have a scalar product, this changes due to the above observation.
This applies, of course, to the tangent space of a manifold with Riemannian metric and induces an isomorphism from the tangent bundle to the cotangent bundle. This is what your exercise is referring to.
(In coordinate systems this is what is usually called lowering and raising indices using the metric tensor, i.e. sending $a_i$ to $g^{ji}a_i$ or $\mu^k$ to $g_{lk} \mu^k$).