Relation between gradient vector and directional derivatives

From Calculus
Jump to: navigation, search

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.