basis for tangent space of manifold

tangent space basis for manifold

Let $M$ be a smooth manifold, $p\in M$ a point, and a chart $\phi :U\subset M⟶V\subset {\mathbb{R}}^n$ such that $p\in U$.

We can define (build) a basis for the tangent space $T_{p}M$ using the standard basis on ${\mathbb{R}}^n$. Define $\frac{\partial }{\partial x_i}{|}_p([f])=D_{e_i}(f\circ {\phi }^{-1})(\phi (p))$ Notes:

• $e_i$ is the standard basis on ${\mathbb{R}}^n$, so we take the directional derivative in that direction.
• $\frac{\partial }{\partial x_i}{|}_p\in T_{p}M$ , so $\frac{\partial }{\partial x_i}{|}_p:{\mathcal{O}}_{M,p}\to \mathbb{R}$
• $f\circ {\phi }^{-1}:{\mathbb{R}}^n\to \mathbb{R}$ so we know how to use multivariable calculus and the total derivative.

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