cotangent space

The cotangent space is the dual space of the tangent space at a point. That’s it.

Basis

Given a basis of the tangent space

$$\left\{{\left(\frac{\partial }{\partial x_1}\right)}_p,{\left(\frac{\partial }{\partial x_2}\right)}_p,\dots ,{\left(\frac{\partial }{\partial x_n}\right)}_p\right\}$$

Then the cotangent space has the dual basis

$$\{(d{x}_1{)}_p,(d{x}_2{)}_p,\dots ,(d{x}_n{)}_p\}$$

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References