Definition of symplectic map I just started reading on symplectic integrator of Hamiltonian system for my physics project, I don't quite understand some of the basic definitions here(see https://na.uni-tuebingen.de/~lubich/chap6.pdf)

Definition 2.1. A linear mapping A : $R^{2d}$ → $R^{2d}$ is called symplectic if $A^{T}JA = J$ or, equivalently, if ω(Aξ, Aη) = ω(ξ, η) for all ξ, η ∈ $R^{2d}$.

where he defined $\omega$ to be the area between vectoors ξ and η. I understand the second definition that says a symplectic linear map preserves the area between any two vectors but how are these two definitions equivalent? 
And the author gave a definition for a (non-linear)differentiable map:

Definition 2.2. A differentiable map g : U → $R^{2d}$
  (where U ⊂ $R^{2d}$ is an open set) is called symplectic if the Jacobian matrix g'(p, q) is everywhere symplectic, i.e., if
  $g'(p, q)^{T}Jg'(p, q) = J$ or ω(g'(p, q)ξ, g'(p, q)η) = ω(ξ, η).

Can someone give me a intuition that why we define symplectic differentiable map like this followed from Definition 2.1?
 A: Well since $\omega(\xi,\eta) = \xi^t J \eta,$ we have $\omega(A\xi,A\eta) = \omega(\xi,\eta)$ iff $\xi^t (A^t J A) \eta = (A\xi)^t J (A\eta) = \xi^t J \eta$ iff the bilinear forms $\omega$ and $\omega'(\xi,\eta) := \xi^t (A^t J A) \eta$ coincide iff their corresponding matrices (namely $J$ and $A^t J A$, respectively) coincide. (Recall that on any $n$-dimensional vector space with a specified basis (here it's $\mathbb{R}^{2d}$ with the standard basis) there is an isomorphism between the space of $(n\times n)$-matrices and the space of bilinear forms given by $A \mapsto ((v,w) \mapsto [v]^t A [w])$ (where the brackets denote the coordinates in the given basis).)

Can someone give me a intuition that why we define symplectic differentiable map like this followed from Definition 2.1?

The most straightforward way to generalize the definition to differentiable maps is to work locally: at any point, the derivative there is a linear map beween $2d$-dimensional vector spaces, so it makes sense to say whether or not it is symplectic. So you can think of such a symplectic differentiable mapping as a map which preserves the symplectic form near every point.
Does that help?
