A symplectic form on a smooth manifold M is a smooth closed 2-form ω on M which is nondegenerate such that at every point m, the alternating bilinear form ω_m on the tangent space T_m M is nondegenerate. A symplectic form on a vector space V over F_q is a function f(x, y) (defined for all x, y element V and taking values in F_q) which satisfies f(λ_1 x_1 + λ_2 x_2, y) = λ_1 f(x_1, y) + λ_2 f(x_2, y) f(y, x) = - f(x, y), and f(x, x) = 0.
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