What is the one form given its value for a vector field?

Question

I read an article on vector fields. the author defined a 1-form on a manifold $M$ as $u(X)=\rho$ when $X$ is a given vector field and $\rho$ is a given real valued function defined on $M$. can we deduce the values of $u(\frac{\partial}{\partial x_{i}}), i=1,n$. I got the following equation \begin{equation} \Sigma \ X^{i} u_{i}=\rho \ , \ u_{i}= u(\frac{\partial}{\partial x_{i}}) \end{equation} It is only one equation in $n$ unknown functions. If I was right is there is a way to get a general form of these function?
When we solve a linear equation in two unknowns we get finitely many solutions and we represent them using a parameter. Can any one give an idea to solve like this one?
The author says that there is a vector field $U$ associated with this form such that $g(U,X)=u(X)$. Can anyone deduce this vector field.
Thanks in advance.

Answer 1

Solution:

The expression $$u(X)=g(U,X)$$ would be an example of the Riesz's representation theorem, the vector $U$ represents the 1-form $u$ in this sense.

By the use of local coordinates we have $U=u^s\partial_s$ and (employing Eisntein's sum convention) then $$u=u_sdx^s,$$ where the relations $$u_k=g_{kt}u^t,$$ are involved. Here $g_{ij}=g(\partial_i, \partial_j)$.