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.