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

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

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

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

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

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