Editing Symplectic manifold (section)

====Example: Cotangent bundle====
The cotangent bundle of a manifold is locally modeled on a space similar to the first example. It can be shown that we can glue these affine symplectic forms hence this bundle forms a symplectic manifold. A less trivial example of a Lagrangian submanifold is the zero section of the cotangent bundle of a manifold. For example, let 

:<math>X = \{(x,y) \in \R^2 : y^2 - x = 0\}.</math>

Then, we can present <math>T^*X</math> as

:<math>T^*X = \{(x,y,\mathrm{d}x,\mathrm{d}y) \in \R^4 : y^2 - x = 0, 2y\mathrm{d}y - \mathrm{d}x = 0\}</math>

where we are treating the symbols <math>\mathrm{d}x,\mathrm{d}y</math> as coordinates of <math>\R^4 = T^*\R^2</math>. We can consider the subset where the coordinates <math>\mathrm{d}x=0</math> and <math>\mathrm{d}y=0</math>, giving us the zero section. This example can be repeated for any manifold defined by the vanishing locus of smooth functions <math>f_1,\dotsc,f_k</math> and their differentials <math>\mathrm{d}f_1,\dotsc,df_k</math>.