Relation between gradient vector and partial derivatives

Statement

Version type	Statement
at a point, in multivariable notation	Suppose $f$ is a real-valued function of $n$ variables $x_{1},x_{2},\dots ,x_{n}$ . Suppose $(a_{1},a_{2},\dots ,a_{n})$ is a point in the domain of $f$ such that the gradient vector of $f$ at $(a_{1},a_{2},\dots ,a_{n})$ , denoted $(\nabla f)(a_{1},a_{2},\dots ,a_{n})$ , exists. Then, the partial derivatives of $f$ with respect to all variables exist, and the coordinates of the gradient vector are the partial derivatives. In other words: $(\nabla f)(a_{1},a_{2},\dots ,a_{n})=\langle f_{x_{1}}(a_{1},a_{2},\dots ,a_{n}),f_{x_{2}}(a_{1},a_{2},\dots ,a_{n}),\dots f_{x_{n}}(a_{1},a_{2},\dots ,a_{n})\rangle$
generic point, in multivariable notation	Suppose $f$ is a real-valued function of $n$ variables $x_{1},x_{2},\dots ,x_{n}$ . Then, we have $(\nabla f)(x_{1},x_{2},\dots ,x_{n})=\langle f_{x_{1}}(x_{1},x_{2},\dots ,x_{n}),f_{x_{2}}(x_{1},x_{2},\dots ,x_{n}),\dots f_{x_{n}}(x_{1},x_{2},\dots ,x_{n})\rangle$ . Equality holds wherever the left side makes sense.
generic point, point-free notation	Suppose $f$ is a function of $n$ variables $x_{1},x_{2},\dots ,x_{n}$ . Then, we have $\nabla f=\langle f_{x_{1}},f_{x_{2}},\dots f_{x_{n}}\rangle$ . Equality holds wherever the left side makes sense.