Editing Cotangent bundle (section)

== Examples ==
The tangent bundle of the vector space <math>\mathbb{R}^n</math> is <math>T\,\mathbb{R}^n = \mathbb{R}^n\times \mathbb{R}^n</math>, and the cotangent bundle is <math>T^*\mathbb{R}^n = \mathbb{R}^n\times (\mathbb{R}^n)^*</math>, where <math>(\mathbb{R}^n)^*</math> denotes the [[dual space]] of covectors, linear functions <math>v^*:\mathbb{R}^n\to \mathbb{R}</math>.

Given a smooth manifold <math>M\subset \mathbb{R}^n</math> embedded as a [[hypersurface]] represented by the vanishing locus of a function <math>f\in C^\infty (\mathbb{R}^n),</math> with the condition that <math>\nabla f \neq 0,</math> the tangent bundle is

:<math>TM = \{(x,v) \in T\,\mathbb{R}^n \ :\ f(x) = 0,\ \, df_x(v) = 0\},</math>

where <math>df_x \in T^*_xM</math> is the [[directional derivative]] <math>df_x(v) = \nabla\! f(x)\cdot v</math>. By definition, the cotangent bundle in this case is

:<math>T^*M = \bigl\{(x,v^*)\in T^*\mathbb{R}^n \ :\ f(x)=0,\ v^* \in T^*_xM \bigr\},</math>
where <math>T^*_xM=\{v \in T_x\mathbb{R}^n\ :\ df_x(v)=0\}^*.</math> Since every covector <math>v^* \in T^*_xM</math> corresponds to a unique vector <math>v \in T_xM</math> for which <math>v^*(u) = v \cdot u,</math> for an arbitrary <math>u \in T_xM,</math>

:<math>T^*M = \bigl\{(x,v^*)\in T^*\mathbb{R}^n\ :\ f(x) = 0,\ v \in T_x\mathbb{R}^n,\ df_x(v)=0 \bigr\}.</math>