Consider the mapping $F:\mathbb R^n\to\mathbb R^m.$ $F=(f^1,f^2,\ldots,f^m)$ is differential at $p\in\mathbb R^n$ iff each $f^i$ is differentiable at $p,$ and $$DF(p)=(Df^1(p),\ldots,Df^m(p)).$$ Here each $Df^i(p)\in L(\mathbb R^n,\mathbb R),$ the space of all linear transformations from $\mathbb R^n$ into $\mathbb R.$ It can be shown that: \begin{align} Df^i(p)(h)&=\lim_{t\to0}\frac{f^i(p+th)-f^i(p)}t\\\\ &=\sum_{j=1}^{n}\frac{\partial f^i}{\partial u^j}(p)h^j,\;i=1,2,\ldots,m,\tag1 \end{align} where $h=\displaystyle\sum_{j=1}^n h^j e_j,$ $(e_j)_{j=1}^n$ being the standard basis of $\mathbb R^n,$ and $u^j$'s are local coordinate functions on $\mathbb R^n.$
In particular, $$Df^i(p)(e_j)=\frac{\partial f^i}{\partial u^j}(p),\;1\le i\le m,\;1\le j\le n.$$ The matrix $\left(\frac{\partial f^i}{\partial u^j}(p)\right)$ is called the Jacobian matrix of $F$ at $p.$
I have taken the above portion from a text on differntial geometry. Here I actually want to know: what exactly is going on in Eq. $(1)?$ Also as indicated, $h$ is a point (or a vector) in $\mathbb R^n.$ So what exactly does the notation $Df^i(p)(h)$ stand for?
Please give some insights...
$Df^i(p)(h)$ is the directional derivative of $f^i$ with starting point $p$ in the direction $h$. $f^i$ itself is the $i$th component of the vector valued function $f$.
(1) is saying that the directional derivative in the direction $h$ at starting point $p$ is given by multiplying the Jacobian at $p$ with the displacement vector. Specifically it says the directional derivative of the scalar function $f^i$ is given by multiplying the gradient of $f^i$ with the displacement vector, which is just the componentwise version of the previous statement.