If $p:M\rightarrow B$ is a surjective submersion and $X$ is a vector field on $M$, where $M$ is a smooth manifold and $B$ is a connected open subset of $\mathbf{R}^2$, that projects to $\partial_x$+$\partial_y$ how to have the part of $X$ that projects to $\partial_x$? If it makes sense.
Thanks for all help me.
Agreeing with Ted Shifrin. Taking the composition of $p$ with the projection $\pi : B \to \Bbb R$, then your question is equivalent to asking what is the preimage of $\partial_x$, which clearly requires the information about derivative to answer. And the whether the preimage is a “part of $X$” is another matter.