Corollary Tangent Vectors are Derivations

Tangent Vectors Are Derivations

Statements

Corollary. Under the conditions of the Theorem Tangent Vectors as Derivations, the canonical basis $\{e_1,\dots ,e_n\}$ for $T_p({\mathbb{R}}^n)$ corresponds to the set $\left\{{\frac{\partial }{\partial x^1}|}_p,\cdots ,{\frac{\partial }{\partial x^n}|}_p\right\}$ of partial derivatives. Thus, for a function $f\in {\mathcal{C}}^{\infty }(p)$, the directional derivative of $f$ in the direction of the vector $v=v^1{e}_1+\cdots +v^n{e}_n$ of $T_p({\mathbb{R}}^n)$ at $p$ can be rewritten as $X_p^{\ast }f={\sum }_{i=1}^n{v}^i{\frac{\partial f}{\partial x^i}|}_p.$

References

@Tu11 pp. 14