Relation between gradient vector and directional derivatives

Statement of relation

Version type	Statement
at a point, in vector notation (multiple variables)	Suppose $f$ is a function of a vector variable $\overline{x} = \langle x_1,x_2,\dots,x_n \rangle$. Suppose $\overline{u}$ is a unit vector and $\overline{a}$ is a point in the domain of $f$. Suppose that the gradient vector of $f$ at $\overline{a}$ exists. We denote this gradient vector by $\nabla f(\overline{a})$. Then, we have the following relationship: $D_{\overline{u}}(f)(\overline{a}) = \overline{u} \cdot (\nabla f(\overline{a}))$ The right side here is the dot product of vectors.
generic point, in vector notation (multiple variables)	Suppose $f$ is a function of a vector variable $\overline{x} = \langle x_1,x_2,\dots,x_n \rangle$. Suppose $\overline{u}$ is a unit vector. We then have: $D_{\overline{u}}(f)(\overline{x}) = \overline{u} \cdot (\nabla f(\overline{x}))$ The right side here is a dot product of vectors. The equality holds whenever the right side makes sense.
generic point, point-free notation (multiple variables)	Suppose $f$ is a function of a vector variable $\overline{x} = \langle x_1,x_2,\dots,x_n \rangle$. Suppose $\overline{u}$ is a unit vector. We then have: $D_{\overline{u}}(f) = \overline{u} \cdot (\nabla f)$ The right side here is a dot product of vector-valued functions (the constant function $\overline{u}$ and the gradient vector of $f$). The equality holds whenever the right side makes sense.