I was studying about gradient and saw the following on Wikipedia:
One is in the introduction:
In coordinate-free terms, the gradient of a function $f(\vec{r})$ may be defined by: $df=\nabla f\bullet d\vec{r} $
Another is in the definition:
The gradient of $f$ is defined as the unique vector field whose dot product with any vector $\vec{v}$ at each point $x$ is the directional derivative of $f$ along $\vec{v}$. That is, $D_vf(x)=(\nabla f(x) \bullet \vec{v}$
I tried to link this two definition because $(\nabla f(x) \bullet \vec{v}$ is similar with $\nabla f \bullet d\vec{r}$.
It is following: $(\nabla f(x) \bullet d\vec{r}=D_{d\vec{r}}f(x)=\lim_{h \rightarrow 0} \frac{f(\vec{x}+hd\vec{r})-f(\vec{x})}{h}$
$\nabla f\bullet d\vec{r} =df=f(\vec{x}+d\vec{r})-f(\vec{x}) $
But i can't still link them.
Can you tell me how these two mean the same thing?