Directional Derivative

Statements

Definition. (@Tu11 pp. 11) Let $F\in {\mathcal{C}}^{\infty }({\mathbb{R}}^n)$, and $c:[0,1]\to {\mathbb{R}}^n$ a curve passing through a point $p=(p^1,\dots ,p^n)$ with direction $v=(v^1,\dots ,v^n)$ given. The directional derivative of $F$ in the direction of $v$ at $p$ is given by the chain rule as $D_{v}F(p)=\frac{F(c(t))-F(p)}{t}={\frac{d}{dt}|}_{t=0}F(c(t))=DF(p){\frac{dc}{dt}(t)|}_{t=0}=DF(p)v.$

References

Based on @Tu11 pp. 11