Definition 1 A Riemannian metric on a smooth manifold is smooth family of inner products on the tangent spaces of M. So g is Riemannian metric if it assigns to each point $p \in M$ a positive definite symmetric bilinear form on $T_p M$,
$$ g_p: T_p M \times T_p M \rightarrow \mathbb{R} $$
with smoothness referring to the requirement that the function
$$ p \mapsto g_p(X_p, Y_p)$$
must be smooth for any locally defined vector fields X,Y in M.
Definition 2 A Riemannian metric on a smooth manifold X is a smooth section of $S^2 T^* X \subset \otimes^2 T^*X$ such that $\left.f\right|_X \in S^2 T^*_xX$ is a positive definite quadratic form on $T_xX$.
Why are these two definitions equivalent?