I'm trying to show that there is a bijection between the set of sections of a smooth vector bundle E \stackrel{\pi}{\to} M and the set of smooth vector bundle morphisms from the trivial line bundle M \times \mathbb{R} \stackrel{\textrm{pr}_M}{\to} M to E \stackrel{\pi}{\to} M.

Since I can't draw a diagram here, the morphisms are given so that \textrm{pr}_M = \tilde{f} \circ \pi (hopefully that makes sense).

Given a section \sigma, I get a morphism by \tilde{f} = \sigma \circ \textrm{pr}_M.

Given a morphism, I get a section by \sigma (p) = \tilde{f} (p,0) \textrm{, } p \in M.

However this does not seem to give a bijection, since sending a morphism to a section and back to a morphism I get \tilde{f} (p,r) = \tilde{f} (p,0) for any r \in \mathbb{R} which doesn't seem right.

Any help would be greatly appreciated, as I have a paper due over Easter.