We choose $(m_{1},\dots,m_{n})$ a basis of $M$ such that $(m_{2},\dots,m_{n})$ is a basis of $M(F)$ and $m_{1}$ points to the exterior direction. Expressing $\lambda$ in this basis we obtain

$$\lambda|_{F}=-\langle v_{F},F\rangle\,\text{\rm d}\operatorname{vol}_{M(F)}.$$

The result then follows from Theorem 3.97. ∎