Exercise 8.9

By Proposition 8.1, the coordinate representations $\hat{X}$ of $X$ and $\hat{Y}$ of $Y$

are smooth in the sense of ordinary calculus. This implies that the coordinate representation

must be smooth in the sense of ordinary calculus. Since our choice of the coordinate chart was arbitrary, the corresponding vector field $\mathit{fX}+\mathit{gY}$ must be smooth.

By Exercise 2.1, $C^{\infty }(M)$ is a commutative ring. Using part (a) of this exercise or using the same strategy (verifying that the coordinate representation satisfies the corresponding smoothness criterion), it is then easy to see that $𝔛(M)$ is a module over $C^{\infty }(M)$ (cf. page 617), i.e.:

(i)

$𝔛(M)$ is an abelian group under addition.

(ii)

Scalar multiplication satisfies $$\begin{array}{llll}\hfill f(\mathit{gX})=(\mathit{fg})X& \text{for all }X\in 𝔛(M)\text{ and }f,g\in C^{\infty }(M);& \hfill & \\ \hfill 1X=X& \text{for all }X\in 𝔛(M).& \hfill & \end{array}$$

(iii)

Scalar multiplication and module addition are related by distributive laws: $$\begin{array}{llll}\hfill (f+g)X=\mathit{fX}+\mathit{gX}& \text{for all }X\in 𝔛(M)\text{ and }f,g\in C^{\infty }(M);& \hfill & \\ \hfill f(X+Y)=\mathit{fX}+\mathit{fY}& \text{for all }X,Y\in 𝔛(M)\text{ and }f\in C^{\infty }(M).& \hfill & \end{array}$$