In the proof of theorem 3.1 they put : $\langle X,\nabla f \rangle =X(f)$ after that they say that: for a curve $c$ on $M$ then $\left\langle\dfrac{\mathrm{d}c}{\mathrm{d}t},\nabla f\right\rangle=\dfrac{\mathrm{d}(f\circ c)}{\mathrm{d}t}$
i don't understand why $<X,\nabla f > =1$ ?
please
Thank you .
$$\dfrac{dc}{dt}=\left(\dfrac{dc^1}{dt}, \dfrac{dc^2}{dt}, \dots , \dfrac{dc^n}{dt}\right)$$ $$\nabla f=\left(\dfrac{\partial f}{\partial x^1}, \dfrac{\partial f}{\partial x^2}, \dots , \dfrac{\partial f}{\partial x^n}\right)$$ Also, we have: $$\dfrac{d(f\circ c)}{dt}=\dfrac{df(c^1(t),c^2(t),\dots ,c^n(t))}{dt}=\dfrac{\partial f}{\partial x^1}\dfrac{dc^1}{dt}+\dots+\dfrac{\partial f}{\partial x^n}\dfrac{dc^n}{dt}=\left\langle\dfrac{dc}{dt},\nabla f\right\rangle$$